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| | Hello, I'm Nicholas, a 22 year old from Pantin, France.<br>My hobbies include (but are not limited to) Computer programming, Audiophilia and watching The Big Bang Theory.<br><br>Here is my webpage ... [http://crazystorebolivia.com/node/296308 Buy Advocare Online] |
| {{Trigonometry}}
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| [[File:Academ Base of trigonometry.svg|thumb|300px|upright=2|Base of trigonometry: if two [[Right triangle|right triangles]] have equal [[Types of angles|acute angles]], they are [[Similarity (geometry)|similar]], so their side lengths [[Proportionality (mathematics)|are proportional]]. Proportionality [[constant (mathematics)|constant]]s are written within the image: sin ''θ'', cos ''θ'', tan ''θ'', where ''θ'' is the common measure of five acute angles.]]
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| In [[mathematics]], the '''trigonometric functions''' (also called the '''circular functions''') are [[function (mathematics)|function]]s of an [[angle]]. They relate the angles of a [[triangle]] to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.
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| The most familiar trigonometric functions are the [[sine]], cosine, and tangent. In the context of the standard unit circle with radius 1 unit, where a triangle is formed by a ray originating at the origin and making some angle with the ''x''-axis, the sine of the angle gives the length of the ''y''-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the ''x''-component (the adjacent of the angle or the run), and the tangent function gives the slope (''y''-component divided by the ''x''-component). More precise definitions are detailed below. Trigonometric functions are commonly defined as [[ratio]]s of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a [[unit circle]]. More modern definitions express them as [[Series (mathematics)|infinite series]] or as solutions of certain [[differential equation]]s, allowing their extension to arbitrary positive and negative values and even to [[complex number]]s.
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| Trigonometric functions have a wide range of uses including computing unknown lengths and angles in [[triangle]]s (often [[right triangle]]s). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a [[vector (mathematics and physics)|vector]] into [[Cartesian coordinate system|Cartesian]] coordinates. The sine and cosine functions are also commonly used to model [[periodic function]] phenomena such as [[sound]] and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.
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| In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the ''definitions'' of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations.
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| ==Right-angled triangle definitions==
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| [[Image:Trigonometry triangle.svg|right|A [[right triangle]] always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.]]
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| [[File:Periodic sine.PNG|thumb|(Top): Trigonometric function sin''θ'' for selected angles ''θ'', ''π'' − ''θ'', ''π'' + ''θ'', and 2''π'' − ''θ'' in the four quadrants. (Bottom) Graph of sine function versus angle. Angles from the top panel are identified.]]
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| The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that [[similarity (geometry)|similar triangles]] maintain the same ratios between their sides. That is, for any similar triangle the ratio of the [[hypotenuse]] (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.
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| To define the trigonometric functions for the angle ''A'', start with any [[right triangle]] that contains the angle ''A''. The three sides of the triangle are named as follows:
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| * The ''hypotenuse'' is the side opposite the right angle, in this case side '''h'''. The hypotenuse is always the longest side of a right-angled triangle.
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| * The ''opposite side'' is the side opposite to the angle we are interested in (angle ''A''), in this case side '''a'''.
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| * The ''adjacent side'' is the side having both the angles of interest (angle ''A'' and right-angle ''C''), in this case side '''b'''.
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| In ordinary [[Euclidean geometry]], according to the [[triangle postulate]], the inside angles of every triangle total 180[[degree (angle)|°]] (π [[radian]]s). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of (0°,90°) as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range. They can be extended to the full set of real arguments by using the [[Unit circle#Trigonometric functions on the unit circle|unit circle]], or by requiring certain symmetries and that they be [[periodic function]]s. For example, the figure shows sin ''θ'' for angles ''θ'', ''π'' − ''θ'', ''π'' + ''θ'', and 2''π'' − ''θ'' depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of ''θ'' is extended to additional rotations, this behavior repeats periodically with a period 2''π''.
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| Rigorously, in [[metric space]], one should express [[angle]], defined as scaled [[arc length]], as a function of triangle sides.
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| It leads to [[inverse trigonometric functions]] first and usual trigonometric functions can be defined by inverting them back.
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| The trigonometric functions are summarized in the following table and described in more detail below. The angle ''θ'' is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram.
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| {| class="wikitable"
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| |-
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| ! Function
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| ! Abbreviation
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| ! Description
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| ! [[List of trigonometric identities|Identities]] (using [[radian]]s)
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| |-
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| ! sine
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| | sin
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| |align=center|opposite / hypotenuse
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| | <math>\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}</math>
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| |-
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| ! cosine
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| | cos
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| |align=center|adjacent / hypotenuse
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| | <math>\cos \theta = \sin \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,</math>
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| |-
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| ! tangent
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| | tan (or tg)
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| |align=center|opposite / adjacent
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| | <math>\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta} </math>
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| |-
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| ! cotangent
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| | cot (or cotan or cotg or ctg or ctn)
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| |align=center|adjacent / opposite
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| | <math>\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta} </math>
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| |-
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| ! secant
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| | sec
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| |align=center|hypotenuse / adjacent
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| | <math>\sec \theta = \csc \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cos \theta} </math>
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| |-
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| ! cosecant
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| | csc (or cosec)
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| |align=center|hypotenuse / opposite
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| | <math>\csc \theta = \sec \left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sin \theta} </math>
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| |}{{clear right}}
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| [[Image:Unitcircledefs.svg|250px|thumb|right|The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle. The number ''θ'' is the length of the curve; thus angles are being measured in [[radian]]s. The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line. ("Fixed" in this context means not moving as ''θ'' changes; "moving" means depending on ''θ''.)
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| Thus, as ''θ'' goes from 0 up to a right angle, sin ''θ'' goes from 0 to 1, tan ''θ'' goes from 0 to ∞, and sec ''θ'' goes from 1 to ∞.]]
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| [[Image:Unitcirclecodefs.svg|250px|thumb|right|The cosine, cotangent, and cosecant functions of an angle ''θ'' constructed geometrically in terms of a unit circle. The functions whose names have the prefix ''co-'' use horizontal lines where the others use vertical lines.]]
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| ===Sine, cosine and tangent {{anchor|sine|cosine|tangent|Sine, cosine and tangent}}===
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| The '''[[sine]]''' of an angle is the ratio of the length of the opposite [[Cathetus|side]] to the length of the hypotenuse. (The word comes from the Latin ''sinus'' for gulf or bay,<ref>Oxford English Dictionary, sine, ''n.''<sup>2</sup></ref> since, given a unit circle, it is the side of the triangle on which the angle ''opens''.) In our case
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| :<math>\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}.</math>
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| Note that this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle ''A'', since all such triangles are [[similarity (geometry)|similar]].
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| {{anchor|cosine}}The '''cosine''' of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: so called because it is the sine of the complementary or co-angle.<ref>Oxford English Dictionary, cosine, ''n.''</ref> In our case
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| :<math>\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}.</math>
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| {{anchor|tangent}}The '''tangent''' of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin ''linea tangens'' or touching line (cf. ''tangere'', to touch).<ref>Oxford English Dictionary, tangent, ''adj. and n.''</ref> In our case
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| :<math>\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}.</math>
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| The acronyms "[[SOH-CAH-TOA|SOHCAHTOA]]" ("Soak-a-toe", "Sock-a-toa", "So-kah-toa") and "[[Trigonometry mnemonics|OHSAHCOAT]]" are commonly used [[mnemonic]]s for these ratios.
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| ===Reciprocal functions===
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| The remaining three functions are best defined using the above three functions.
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| The '''cosecant''' csc(''A''), or cosec(''A''), is the [[Multiplicative inverse|reciprocal]] of sin(''A''); i.e. the ratio of the length of the hypotenuse to the length of the opposite side:
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| :<math>\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}. </math>
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| The '''secant''' sec(''A'') is the [[multiplicative inverse|reciprocal]] of cos(''A''); i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:
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| :<math>\sec A = \frac {1}{\cos A} = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}. </math>
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| It is so called because it represents the line that ''cuts'' the circle (from Latin: ''secare'', to cut).<ref>Oxford English Dictionary, secant, ''adj.'' and ''n.''</ref>
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| The '''cotangent''' cot(''A'') is the [[multiplicative inverse|reciprocal]] of tan(''A''); i.e. the ratio of the length of the adjacent side to the length of the opposite side:
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| :<math>\cot A = \frac {1}{\tan A} = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}. </math>
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| ===Slope definitions===
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| Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the ''rise'', ''run'', and ''[[slope]]'' of a line segment relative to horizontal. The slope is commonly taught as "rise over run" or {{math|{{frac|rise|run}}}}. The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a line segment length of 1 (as in a [[unit circle]]), the following [[mnemonic device]]s show the correspondence of definitions:
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| # "Sine is first, rise is first" meaning that Sine takes the angle of the line segment and tells its vertical rise when the length of the line is 1.
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| # "Cosine is second, run is second" meaning that Cosine takes the angle of the line segment and tells its horizontal run when the length of the line is 1.
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| # "Tangent combines the rise and run" meaning that Tangent takes the angle of the line segment and tells its slope; or alternatively, tells the vertical rise when the line segment's horizontal run is 1.
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| This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, ''i.e.,'' angles and slopes. (Note that the arctangent or "inverse tangent" is not to be confused with the ''cotangent,'' which is cosine divided by sine.)
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| While the length of the line segment makes no difference for the slope (the slope does not depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run when the line does not have a length of 1, just multiply the sine and cosine by the line length. For instance, if the line segment has length 5, the run at an angle of 7° is 5 cos(7°)
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| ==Unit-circle definitions==
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| [[Image:Unit circle angles color.svg|right|thumb|300px|The [[unit circle]]]]
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| The six trigonometric functions can also be defined in terms of the [[unit circle]], the [[circle]] of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles.
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| The unit circle definition does, however, permit the definition of the trigonometric functions for all positive and negative arguments, not just for angles between 0 and π/2 [[radian]]s.
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| It also provides a single visual picture that encapsulates at once all the important triangles. From the [[Pythagorean theorem]] the equation for the unit circle is:
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| : <math>x^2 + y^2 = 1. \, </math>
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| In the picture, some common angles, measured in radians, are given. Measurements in the counterclockwise direction are positive angles and measurements in the clockwise direction are negative angles.
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| Let a line through the origin, making an angle of ''θ'' with the positive half of the ''x''-axis, intersect the unit circle. The ''x''- and ''y''-coordinates of this point of intersection are equal to cos ''θ'' and sin ''θ'', respectively.
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| The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin ''θ'' = ''y''/1 and cos ''θ'' = ''x''/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.
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| Note that these values (sin 0°, sin 30°, sin 45°, sin 60° and sin 90°) can easily be memorized in the form
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| : <math>\frac{1}{2}\sqrt{0},\quad \frac{1}{2}\sqrt{1},\quad \frac{1}{2}\sqrt{2},\quad \frac{1}{2}\sqrt{3},\quad \frac{1}{2}\sqrt{4},</math>
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| but the angles are not equally spaced.
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| The values for 15°, 18°, 36°, 54°, 72°, and 75° are derived as follows:
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| :<math>\sin 15^\circ = \cos 75^\circ = \dfrac{\sqrt6 - \sqrt2}{4}\,\!</math>
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| :<math>\sin 18^\circ = \cos 72^\circ = \frac{\sqrt5 - 1}{4}</math>
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| :<math>\sin 36^\circ = \cos 54^\circ = \frac{\sqrt{10 - 2\sqrt5}}{4}</math>
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| :<math>\sin 54^\circ = \cos 36^\circ = \dfrac{\sqrt5 + 1 } {4}\,\!</math>
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| :<math>\sin 72^\circ = \cos 18^\circ = \frac{\sqrt{10 + 2\sqrt5}}{4}</math>
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| :<math>\sin 75^\circ = \cos 15^\circ = \dfrac{\sqrt6 + \sqrt2}{4}.\,\!</math>
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| From these, the values for all multiples of 3° can be analytically computed. For example:
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| :<math>\sin 3^\circ = \cos 87^\circ = \dfrac{\sqrt{30} + \sqrt{10} + \sqrt{20 + 4 \sqrt5} - \sqrt6 - \sqrt2 - \sqrt{60 + 12 \sqrt5}}{16}\,\!</math>
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| :<math>\sin 6^\circ = \cos 84^\circ = \dfrac{\sqrt{30 - 6 \sqrt5} - \sqrt5 - 1 }{8}\,\!</math>
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| :<math>\sin 9^\circ = \cos 81^\circ = \dfrac{\sqrt{90} + \sqrt{18} + \sqrt{10} + \sqrt2 - \sqrt{20 - 4 \sqrt5} - \sqrt{180 - 36 \sqrt5}}{32}\,\!</math>
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| :<math>\sin 84^\circ = \cos 6^\circ =\frac{\sqrt{10 - 2\sqrt5} + \sqrt{15} + \sqrt3}{8}</math>
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| :<math>\sin 87^\circ = \cos 3^\circ =\frac{\sqrt{60 + 12\sqrt5}+\sqrt{20 + 4\sqrt5}+\sqrt{30}+\sqrt2-\sqrt6-\sqrt{10}}{16}</math>
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| Though a complex task, the analytical expression of sin 1° can be obtained by analytically solving the [[cubic functions | cubic equation]]
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| :<math>\sin 3^\circ = 3\sin 1^\circ - 4\sin^3 1^\circ</math>
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| from whose solution one can analytically derive trigonometric functions of all angles of integer degrees.
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| {{clr}}
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| [[Image:Sine cosine one period.svg|300px|right|thumb|The sine and cosine functions graphed on the Cartesian plane.]]
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| For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine and cosine are [[periodic function]]s with period 2π:
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| : <math>\sin\theta = \sin\left(\theta + 2\pi k \right),\,</math>
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| : <math>\cos\theta = \cos\left(\theta + 2\pi k \right),\,</math>
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| for any angle θ and any [[integer]] ''k''.
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| The ''smallest'' positive period of a periodic function is called the ''primitive period'' of the function.
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| The primitive period of the sine or cosine is a full circle, i.e. 2π radians or 360 degrees.
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| {{clr}}
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| Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:
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| : <math>
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| \begin{align}
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| \tan\theta & = \frac{\sin\theta}{\cos\theta},\ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta} \\
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| \sec\theta & = \frac{1}{\cos\theta},\ \csc\theta = \frac{1}{\sin\theta}
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| \end{align}
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| </math>
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| So :
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| * The primitive period of the secant, or cosecant is also a full circle, i.e. 2π radians or 360 degrees.
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| * The primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.
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| [[Image:Trigonometric functions.svg|right|thumb|300px|Trigonometric functions:
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| <span style="color:#00A">Sine</span>,
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| <span style="color:#0A0">Cosine</span>,
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| <span style="color:#A00">Tangent</span>,
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| <span style="color:#00A">Cosecant (dotted)</span>,
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| <span style="color:#0A0">Secant (dotted)</span>,
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| <span style="color:#A00">Cotangent (dotted)</span>]]
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| The image at right includes a graph of the tangent function.
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| * Its ''θ''-intercepts correspond to those of sin(''θ'') while its undefined values correspond to the ''θ''-intercepts of cos(''θ'').
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| * The function changes slowly around angles of ''k''π, but changes rapidly at angles close to (''k'' + 1/2)π.
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| * The graph of the tangent function also has a vertical [[asymptote]] at ''θ'' = (''k'' + 1/2)π, the ''θ''-intercepts of the cosine function, because the function approaches infinity as ''θ'' approaches (''k'' + 1/2)π from the left and minus infinity as it approaches (''k'' + 1/2)π from the right.
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| {{clr}}
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| [[Image:Circle-trig6.svg|right|thumb|300px|All of the trigonometric functions of the angle ''θ'' can be constructed geometrically in terms of a unit circle centered at ''O''.]]
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| Alternatively, ''all'' of the basic trigonometric functions can be defined in terms of a unit circle centered at ''O'' (as shown in the picture to the right), and similar such geometric definitions were used historically.
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| * In particular, for a chord ''AB'' of the circle, where ''θ'' is half of the subtended angle, sin(''θ'') is ''AC'' (half of the chord), a definition introduced in [[India]]<ref name=boyer/> (see [[history of trigonometry|history]]).
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| * cos(''θ'') is the horizontal distance ''OC'', and [[versine|versin]](''θ'') = 1 − cos(''θ'') is ''CD''.
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| * tan(''θ'') is the length of the segment ''AE'' of the tangent line through ''A'', hence the word ''[[tangent]]'' for this function. cot(''θ'') is another tangent segment, ''AF''.
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| * sec(''θ'') = ''OE'' and csc(''θ'') = ''OF'' are segments of [[secant line]]s (intersecting the circle at two points), and can also be viewed as projections of ''OA'' along the tangent at ''A'' to the horizontal and vertical axes, respectively.
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| * ''DE'' is [[exsecant|exsec]](''θ'') = sec(''θ'') − 1 (the portion of the secant outside, or ''ex'', the circle).
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| * From these constructions, it is easy to see that the secant and tangent functions diverge as ''θ'' approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as ''θ'' approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.<ref>See Maor (1998)</ref>)
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| {{clr}}
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| ==Series definitions==
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| [[Image:Taylorsine.svg|300px|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]
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| Trigonometric functions are [[analytic functions]]. Using only geometry and properties of [[limit of a function|limits]], it can be shown that the [[derivative]] of sine is cosine and the derivative of cosine is the negative of sine. (Here, and generally in [[calculus]], all angles are measured in [[radian]]s; see also [[#The significance of radians|the significance of radians]] below.) One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s ''x'':<ref>See Ahlfors, pages 43–44.</ref>
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| :<math>
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| \begin{align}
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| \sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\
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| & = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \\
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| \cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\
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| & = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}.
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| \end{align}
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| </math>
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| These identities are sometimes taken as the ''definitions'' of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g., in [[Fourier series]]), since the theory of [[Series (mathematics)|infinite series]] can be developed, independent of any geometric considerations, from the foundations of the [[real number|real number system]]. The [[derivative|differentiability]] and [[continuous function|continuity]] of these functions are then established from the series definitions alone. The value of [[pi|{{pi}}]] can be defined as the smallest positive number for which sin = 0.
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| Other series can be found.<ref>Abramowitz; Weisstein.</ref> For the following trigonometric functions:
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| : ''U''<sub>''n''</sub> is the ''n''th [[up/down number]],
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| : ''B''<sub>''n''</sub> is the ''n''th [[Bernoulli number]], and
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| : ''E''<sub>''n''</sub> (below) is the ''n''th [[Euler number]].
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| <!-- ''It would be nice if someone adds how the infinite series expressions are deduced.'' -->
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| <!-- Yeah but those numbers are often _defined_ by the power series. -->
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| '''Tangent'''
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| : <math>
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| \begin{align}
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| \tan x & {} = \sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!} \\
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| & {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} x^{2n-1}}{(2n)!} \\
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| & {} = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + \frac{17}{315}x^7 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
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| \end{align}
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| </math>
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| When this series for the tangent function is expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "tangent numbers", have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets of odd cardinality.<ref>Stanley, Enumerative Combinatorics, Vol I., page 149</ref>
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| '''Cosecant'''
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| : <math>
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| \begin{align}
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| \csc x & {} = \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 (2^{2n-1}-1) B_{2n} x^{2n-1}}{(2n)!} \\
| |
| & {} = x^{-1} + \frac{1}{6}x + \frac{7}{360}x^3 + \frac{31}{15120}x^5 + \cdots, \qquad \text{for } 0 < |x| < \pi.
| |
| \end{align}
| |
| </math>
| |
| | |
| <!-- extra blank line between [[TeX]] displays for legibility -->
| |
| '''Secant'''
| |
| : <math>
| |
| \begin{align}
| |
| \sec x & {} = \sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!}
| |
| = \sum_{n=0}^\infty \frac{(-1)^n E_{2n} x^{2n}}{(2n)!} \\
| |
| & {} = 1 + \frac{1}{2}x^2 + \frac{5}{24}x^4 + \frac{61}{720}x^6 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
| |
| \end{align}
| |
| </math>
| |
| | |
| When this series for the secant function is expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "secant numbers", have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets of even cardinality.<ref>Stanley, Enumerative Combinatorics, Vol I</ref>
| |
| | |
| '''Cotangent'''
| |
| : <math>
| |
| \begin{align}
| |
| \cot x & {} = \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!} \\
| |
| & {} = x^{-1} - \frac{1}{3}x - \frac{1}{45}x^3 - \frac{2}{945}x^5 - \cdots, \qquad \text{for } 0 < |x| < \pi.
| |
| \end{align}
| |
| </math>
| |
| | |
| From a theorem in [[complex analysis]], there is a unique [[analytic continuation]] of this real function to the domain of complex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above.
| |
| | |
| There is a series representation as [[partial fraction expansion]] where just translated [[Multiplicative inverse|reciprocal function]]s are summed up, such that the [[Pole (complex analysis)|pole]]s of the cotangent function and the reciprocal functions match:<ref>
| |
| {{cite book
| |
| | last1 = Aigner
| |
| | first1 = Martin
| |
| | author1-link = Martin Aigner
| |
| | last2 = Ziegler
| |
| | first2 = Günter M.
| |
| | author2-link = Günter Ziegler
| |
| | title = Proofs from THE BOOK
| |
| | publisher = [[Springer-Verlag]]
| |
| | edition = Second
| |
| | year = 2000
| |
| | isbn = 978-3-642-00855-9
| |
| | page=149
| |
| | url = http://www.springer.com/mathematics/book/978-3-642-00855-9
| |
| }}</ref>
| |
| : <math>
| |
| \pi \cdot \cot (\pi x) = \lim_{N\to\infty}\sum_{n=-N}^N \frac{1}{x+n}.
| |
| </math>
| |
| This identity can be proven with the [[Gustav Herglotz|Herglotz]] trick.<ref>{{cite book
| |
| |title=Theory of complex functions
| |
| |first1=Reinhold
| |
| |last1=Remmert
| |
| |publisher=Springer
| |
| |year=1991
| |
| |isbn=0-387-97195-5
| |
| |page=327
| |
| |url=http://books.google.com/books?id=CC0dQxtYb6kC}}, [http://books.google.com/books?id=CC0dQxtYb6kC&pg=PA327 Extract of page 327]
| |
| </ref>
| |
| By combining the <math>-n</math>-th with the <math>n</math>-th term, it can be expressed as an [[absolute convergence|absolutely convergent]] series:
| |
| :<math>
| |
| \pi \cdot \cot (\pi x) = \frac{1}{x} + \sum_{n=1}^\infty \frac{2x}{x^2-n^2}.
| |
| </math>
| |
| | |
| ===Relationship to exponential function and complex numbers===
| |
| It can be shown from the series definitions<ref>For a demonstration, see [[Euler's_formula#Using_power_series]]</ref> that the sine and cosine functions are the [[complex number|imaginary]] and real parts, respectively, of the [[exponential function#On the complex plane|complex exponential function]] when its argument is purely imaginary:
| |
| | |
| :<math> e^{i\theta} = \cos\theta + i\sin\theta. \, </math>
| |
| | |
| This identity is called [[Euler's formula]]. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the [[complex plane]], parametrized by ''e''<sup> ''ix''</sup>, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.
| |
| | |
| [[Euler's formula]] can also be used to derive some [[List of trigonometric identities|trigonometric identities]], by writing sine and cosine as:
| |
| | |
| : <math> \sin\theta = \frac{e^{i \theta} - e^{-i \theta}}{2i} \;</math>
| |
| | |
| : <math> \cos\theta = \frac{e^{i \theta} + e^{-i \theta}}{2} \;</math>
| |
| | |
| Furthermore, this allows for the definition of the trigonometric functions for complex arguments ''z'':
| |
| | |
| : <math>\sin z = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} = \frac{e^{i z} - e^{-i z}}{2i}\, = \frac{\sinh \left( i z\right) }{i} </math>
| |
| | |
| : <math>\cos z = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n)!}z^{2n} = \frac{e^{i z} + e^{-i z}}{2}\, = \cosh \left(i z\right) </math>
| |
| | |
| where ''i''<sup> 2</sup> = −1. The sine and cosine defined by this are [[entire function]]s. Also, for purely real ''x'',
| |
| | |
| : <math>\cos x = \operatorname{Re}(e^{i x}) \,</math>
| |
| | |
| : <math>\sin x = \operatorname{Im}(e^{i x}) \,</math>
| |
| | |
| It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments.
| |
| | |
| : <math>\sin (x + iy) = \sin x \cosh y + i \cos x \sinh y,\,</math>
| |
| | |
| : <math>\cos (x + iy) = \cos x \cosh y - i \sin x \sinh y.\,</math>
| |
| | |
| This exhibits a deep relationship between the complex sine and cosine functions and their real (''sin'', ''cos'') and hyperbolic real (''sinh'', ''cosh'') counterparts.
| |
| | |
| ====Complex graphs====
| |
| In the following graphs, the domain is the complex plane pictured, and the range values are indicated at each point by color. Brightness indicates the size (absolute value) of the range value, with black being zero. Hue varies with argument, or angle, measured from the positive real axis. ([[:Image:Complex coloring.jpg|more]])
| |
| | |
| {| style="text-align:center"
| |
| |+ '''Trigonometric functions in the complex plane'''
| |
| |[[Image:Complex sin.jpg|1000x136px|none]]
| |
| |[[Image:Complex cos.jpg|1000x136px|none]]
| |
| |[[Image:Complex tan.jpg|1000x136px|none]]
| |
| |[[Image:Complex Cot.jpg|1000x136px|none]]
| |
| |[[Image:Complex Sec.jpg|1000x136px|none]]
| |
| |[[Image:Complex Csc.jpg|1000x136px|none]]
| |
| |-
| |
| |<math>
| |
| \sin z\,
| |
| </math>
| |
| |<math>
| |
| \cos z\,
| |
| </math>
| |
| |<math>
| |
| \tan z\,
| |
| </math>
| |
| |<math>
| |
| \cot z\,
| |
| </math>
| |
| |<math>
| |
| \sec z\,
| |
| </math>
| |
| |<math>
| |
| \csc z\,
| |
| </math>
| |
| |}
| |
| | |
| ==Definitions via differential equations==
| |
| Both the sine and cosine functions satisfy the [[differential equation]]:
| |
| : <math>y'' = -y.\,</math>
| |
| | |
| That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional [[function space]] ''V'' consisting of all solutions of this equation,
| |
| * the sine function is the unique solution satisfying the initial condition <math>\scriptstyle \left( y'(0), y(0) \right) = (1, 0)\,</math> and
| |
| * the cosine function is the unique solution satisfying the initial condition <math>\scriptstyle \left( y'(0), y(0) \right) = (0, 1)\,</math>.
| |
| | |
| Since the sine and cosine functions are linearly independent, together they form a [[basis (linear algebra)|basis]] of ''V''. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See [[linear differential equation]].) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the [[List of trigonometric identities|trigonometric identities]] for the sine and cosine functions.
| |
| | |
| Further, the observation that sine and cosine satisfies ''y''′′ = −''y'' means that they are [[eigenfunction]]s of the second-derivative operator.
| |
| | |
| The tangent function is the unique solution of the nonlinear differential equation
| |
| : <math>y' = 1 + y^2\,</math>
| |
| satisfying the initial condition ''y''(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation.<ref>{{cite book|isbn= 0-19-853446-9|title=Visual Complex Analysis|last=Needham|first=p}}</ref>
| |
| | |
| ===The significance of radians===
| |
| Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them. If an argument to sine or cosine in radians is scaled by frequency,
| |
| | |
| : <math>f(x) = \sin kx, \,</math>
| |
| | |
| then the derivatives will scale by ''amplitude''.
| |
| | |
| : <math>f'(x) = k\cos kx. \,</math>
| |
| | |
| Here, ''k'' is a constant that represents a mapping between units. If ''x'' is in degrees, then | |
| | |
| : <math>k = \frac{\pi}{\textstyle 180^\circ}.</math>
| |
| | |
| This means that the second derivative of a sine in degrees does not satisfy the differential equation
| |
| | |
| : <math>y'' = -y\,</math>
| |
| | |
| but rather
| |
| | |
| : <math>y'' = -k^2 y.\,</math>
| |
| | |
| The cosine's second derivative behaves similarly.
| |
| | |
| This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.
| |
| | |
| ==Identities==
| |
| {{Main|List of trigonometric identities}}
| |
| Many identities interrelate the trigonometric functions. Among the most frequently used is the '''Pythagorean identity,''' which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the [[Pythagorean theorem]]. In symbolic form, the Pythagorean identity is written
| |
| | |
| :<math>\sin^2 x + \cos^2 x = 1 </math>
| |
| | |
| where <math>\textstyle \sin^2 x + \cos^2 x = 1 </math> is standard notation for <math>\textstyle (\sin x)^2 + (\cos x)^2 = 1 </math>
| |
| | |
| Other key relationships are the '''sum and difference formulas,''' which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments that date to [[Ptolemy]]. One can also produce them algebraically using [[Euler's formula]].
| |
| ; Sum
| |
| :<math>\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y, \,</math>
| |
| :<math>\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y, \,</math>
| |
| ; Subtraction
| |
| :<math>\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y, \,</math>
| |
| :<math>\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y. \,</math>
| |
| | |
| These in turn lead to the following three-angle formulae:
| |
| | |
| :<math>\sin \left(x+y+z\right)=\sin x \cos y \cos z + \sin y \cos z \cos x + \sin z \cos y \cos x - \sin x \sin y \sin z, \,</math>
| |
| :<math>\cos \left(x+y+z\right)=\cos x \cos y \cos z - \cos x \sin y \sin z - \cos y \sin x \sin z - \cos z \sin x \sin y, \,</math>
| |
| | |
| When the two angles are equal, the sum formulas reduce to simpler equations known as the '''double-angle formulae.'''
| |
| | |
| :<math>\sin 2x = 2 \sin x \cos x, \,</math>
| |
| :<math>\cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x. \,</math>
| |
| | |
| When three angles are equal, the three-angle formulae simplify to
| |
| :<math>\sin 3x = 3 \sin x - 4 \sin^3 x . \,</math>
| |
| :<math>\cos 3x = 4 \cos^3 x - 3 \cos x . \,</math>
| |
| | |
| These identities can also be used to derive the [[product-to-sum identities]] that were used in antiquity to transform the product of two numbers into a sum of numbers and greatly speed operations, much like the [[logarithm|logarithm function]].
| |
| | |
| ===Calculus===
| |
| For [[integral]]s and [[derivative]]s of trigonometric functions, see the relevant sections of [[Differentiation of trigonometric functions]], [[Lists of integrals]] and [[List of integrals of trigonometric functions]]. Below is the list of the derivatives and integrals of the six basic trigonometric functions. The number ''C'' is a constant of integration.
| |
| | |
| :{| class="wikitable"
| |
| |-
| |
| | <math>\ \ \ \ f(x)</math>
| |
| | <math>\ \ \ \ f'(x)</math>
| |
| | <math>\int f(x)\,dx</math>
| |
| |-
| |
| | <math>\,\ \sin x</math>
| |
| | <math>\,\ \cos x</math>
| |
| | <math>\,\ -\cos x + C</math>
| |
| |-
| |
| | <math>\,\ \cos x</math>
| |
| | <math>\,\ -\sin x</math>
| |
| | <math>\,\ \sin x + C</math>
| |
| |-
| |
| | <math>\,\ \tan x</math>
| |
| | <math>\,\ \sec^2 x = 1+\tan^2 x</math>
| |
| | <math>-\ln \left |\cos x\right | + C</math>
| |
| |-
| |
| | <math>\,\ \cot x</math>
| |
| | <math>\,\ -\csc^2 x = -(1+\cot^2 x)</math>
| |
| | <math>\ln \left |\sin x\right | + C</math>
| |
| |-
| |
| | <math>\,\ \sec x</math>
| |
| | <math>\,\ \sec x\tan x</math>
| |
| | <math>\ln \left |\sec x + \tan x\right | + C</math>
| |
| |-
| |
| | <math>\,\ \csc x</math>
| |
| | <math>\,\ -\csc x \cot x</math>
| |
| | <math>\ -\ln \left |\csc x + \cot x\right | + C</math>
| |
| |}
| |
| | |
| ===Definitions using functional equations===
| |
| In [[mathematical analysis]], one can define the trigonometric functions using [[functional equation]]s based on properties like the difference formula. Taking as given these formulas, one can prove that only two [[real analysis|real function]]s satisfy those conditions. Symbolically, we say that there exists exactly one pair of real functions — <math>\scriptstyle \sin\,</math> and <math>\scriptstyle \cos\,</math> — such that for all real numbers <math>\scriptstyle x\,</math> and <math>\scriptstyle y\,</math>, the following equation hold:<ref name=kannappan>{{cite book|last=Kannappan|first=Palaniappan|title=Functional Equations and Inequalities with Applications|year=2009|publisher=Springer|isbn=978-0387894911}}</ref>
| |
| | |
| : <math>\cos(x- y) = \cos x\cos y + \sin x\sin y\,</math>
| |
| | |
| with the added condition that
| |
| | |
| : <math>0 < x\cos x < \sin x < x\hbox{ for }0 < x < 1.\,</math>
| |
| | |
| Other derivations, starting from other functional equations, are also possible, and such derivations can be extended to the complex numbers.
| |
| As an example, this derivation can be used to define [[trigonometry in Galois fields]].
| |
| | |
| ==Computation==
| |
| The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of [[computer]]s and [[scientific calculator]]s that provide built-in trigonometric functions for any angle. This section, however, describes details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found.
| |
| | |
| The first step in computing any trigonometric function is range reduction—reducing the given angle to a "reduced angle" inside a small range of angles, say 0 to ''π''/2, using the periodicity and symmetries of the trigonometric functions.
| |
| | |
| {{Main|Generating trigonometric tables}}
| |
| | |
| Prior to computers, people typically evaluated trigonometric functions by [[interpolation|interpolating]] from a detailed table of their values, calculated to many [[significant figures]]. Such tables have been available for as long as trigonometric functions have been described (see [[#History|History]] below), and were typically generated by repeated application of the half-angle and angle-addition [[List of trigonometric identities|identities]] starting from a known value (such as sin(''π''/2) = 1).
| |
| | |
| Modern computers use a variety of techniques.<ref>Kantabutra.</ref> One common method, especially on higher-end processors with [[floating point]] units, is to combine a [[polynomial]] or [[rational function|rational]] [[approximation theory|approximation]] (such as [[Approximation theory|Chebyshev approximation]], best uniform approximation, and [[Padé approximant|Padé approximation]], and typically for higher or variable precisions, [[Taylor series|Taylor]] and [[Laurent series]]) with range reduction and a [[lookup table|table lookup]]—they first look up the closest angle in a small table, and then use the polynomial to compute the correction.<ref>However, doing that while maintaining precision is nontrivial, and methods like [[Gal's accurate tables]], Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used.</ref> Devices that lack [[Arithmetic logic unit|hardware multipliers]] often use an algorithm called [[CORDIC]] (as well as related techniques), which uses only addition, subtraction, [[bitwise operation|bitshift]], and [[lookup table|table lookup]]. These methods are commonly implemented in [[Personal computer hardware|hardware]] [[floating-point unit]]s for performance reasons.
| |
| | |
| For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the [[arithmetic-geometric mean]], which itself approximates the trigonometric function by the ([[complex number|complex]]) [[elliptic integral]].<ref>{{cite web|url=http://doi.acm.org/10.1145/321941.321944|title=R. P. Brent, "Fast Multiple-Precision Evaluation of Elementary Functions", J. ACM '''23''', 242 (1976).}}</ref>
| |
| | |
| {{Main|Exact trigonometric constants}}
| |
| | |
| Finally, for some simple angles, the values can be easily computed by hand using the [[Pythagorean theorem]], as in the following examples. For example, the sine, cosine and tangent of any integer multiple of <math>\pi / 60</math> [[radian]]s (3°) can be found [[Exact trigonometric constants|exactly by hand]].
| |
| | |
| Consider a right triangle where the two other angles are equal, and therefore are both <math>\pi / 4</math> radians (45°). Then the length of side ''b'' and the length of side ''a'' are equal; we can choose <math>a = b = 1</math>. The values of sine, cosine and tangent of an angle of <math>\pi / 4</math> radians (45°) can then be found using the Pythagorean theorem:
| |
| | |
| :<math>c = \sqrt { a^2+b^2 } = \sqrt2\,.</math>
| |
| | |
| Therefore:
| |
| | |
| :<math>\sin \frac{\pi}{4} = \sin 45^\circ = \cos \frac{\pi}{4}= \cos 45^\circ = {1 \over \sqrt2},\,</math>
| |
| | |
| :<math>\tan \frac{\pi}{4} = \tan 45^\circ = { {\sin \frac{\pi}{4} } \over {\cos \frac{\pi}{4} } }= {1 \over \sqrt2} \cdot {\sqrt2 \over 1} = {\sqrt2 \over \sqrt2} = 1. \,</math>
| |
| | |
| To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, the shortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:
| |
| | |
| :<math>\sin \frac{\pi}{6} = \sin 30^\circ = \cos \frac{\pi}{3} = \cos 60^\circ = {1 \over 2}\,,</math>
| |
| :<math>\cos \frac{\pi}{6} = \cos 30^\circ = \sin \frac{\pi}{3} = \sin 60^\circ = {\sqrt3 \over 2}\,,</math>
| |
| :<math>\tan \frac{\pi}{6} = \tan 30^\circ = \cot \frac{\pi}{3} = \cot 60^\circ = {1 \over \sqrt3}\,.</math>
| |
| <!-- Please do not expand this section by adding more and more formulas for different angles, or more formulas for the same angles. We have a whole article on [[Exact trigonometric constants]], linked above. Go expand that. -->
| |
| | |
| ===Special values in trigonometric functions===
| |
| There are some commonly used special values in trigonometric functions, as shown in the following table.
| |
| | |
| {| class="wikitable" style="text-align: center;"
| |
| |-
| |
| ! Function
| |
| ! <math>0 \ (0^\circ)</math>
| |
| ! <math>\frac{\pi}{12} \ (15^\circ)</math>
| |
| ! <math>\frac{\pi}{8} \ (22.5^\circ)</math>
| |
| ! <math>\frac{\pi}{6} \ (30^\circ)</math>
| |
| ! <math>\frac{\pi}{4} \ (45^\circ)</math>
| |
| ! <math>\frac{\pi}{3} \ (60^\circ)</math>
| |
| ! <math>\frac{5\pi}{12} \ (75^\circ)</math>
| |
| !<math>\frac{\pi}{2} \ (90^\circ)</math>
| |
| |-
| |
| | sin
| |
| | <math>0</math>
| |
| | <math>\frac{ \sqrt{6} - \sqrt{2} } {4}</math>
| |
| | <math>\frac{ \sqrt{2 - \sqrt{2}} } {2}</math>
| |
| | <math>\frac{1}{2}</math>
| |
| | <math>\frac{\sqrt{2}}{2}</math>
| |
| | <math>\frac{\sqrt{3}}{2}</math>
| |
| | <math>\frac{ \sqrt{6} + \sqrt{2} } {4}</math>
| |
| | <math>1</math>
| |
| |-
| |
| | cos
| |
| | <math>1</math>
| |
| | <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| |
| | <math>\frac{ \sqrt{2 + \sqrt{2}} } {2}</math>
| |
| | <math>\frac{\sqrt{3}}{2}</math>
| |
| | <math>\frac{\sqrt{2}}{2}</math>
| |
| | <math>\frac{1}{2}</math>
| |
| | <math>\frac{ \sqrt{6} - \sqrt{2}} {4}</math>
| |
| | <math>0</math>
| |
| |-
| |
| | tan
| |
| | <math>0</math>
| |
| | <math>2-\sqrt{3}</math>
| |
| | <math>\sqrt{2} - 1</math>
| |
| | <math>\frac{1}{\sqrt{3}}</math>
| |
| | <math>1</math>
| |
| | <math>\sqrt{3}</math>
| |
| | <math>2+\sqrt{3}</math>
| |
| | <math>\infty</math><ref name="Abramowitz and Stegun">Abramowitz, Milton and Irene A. Stegun, p.74</ref>
| |
| |-
| |
| | cot
| |
| | <math>\infty</math><ref name="Abramowitz and Stegun"/>
| |
| | <math>2+\sqrt{3}</math>
| |
| | <math>\sqrt{2} + 1</math>
| |
| | <math>\sqrt{3}</math>
| |
| | <math>1</math>
| |
| | <math>\frac{\sqrt{3}}{3}</math>
| |
| | <math>2-\sqrt{3}</math>
| |
| | <math>0</math>
| |
| |-
| |
| | sec
| |
| | <math>1</math>
| |
| | <math>\sqrt{6} - \sqrt{2}</math>
| |
| | <math>\sqrt{2} \sqrt{ 2 - \sqrt{2} }</math>
| |
| | <math>\frac{2}{\sqrt{3}}</math>
| |
| | <math>\sqrt{2}</math>
| |
| | <math>2</math>
| |
| | <math>\sqrt{6}+\sqrt{2}</math>
| |
| | <math>\infty</math><ref name="Abramowitz and Stegun"/>
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| |-
| |
| | csc
| |
| | <math>\infty</math><ref name="Abramowitz and Stegun"/>
| |
| | <math>\sqrt{6}+\sqrt{2}</math>
| |
| | <math>\sqrt{2} \sqrt{ 2 + \sqrt{2} }</math>
| |
| | <math>2</math>
| |
| | <math>\sqrt{2}</math>
| |
| | <math>\frac{2\sqrt{3}}{3}</math>
| |
| | <math>\sqrt{6} - \sqrt{2}</math>
| |
| | <math>1</math>
| |
| |}
| |
| The symbol <math>\infty</math> here represents the [[point at infinity]] on the [[real projective line]], the limit on the [[extended real line]] is <math>+\infty</math> on one side and <math>-\infty</math> on the other.
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| | |
| ==Inverse functions==
| |
| {{Main|Inverse trigonometric functions}}
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| The trigonometric functions are periodic, and hence not [[injective function|injective]], so strictly they do not have an [[inverse function]]. Therefore to define an inverse function we must restrict their domains so that the trigonometric function is [[bijection|bijective]]. In the following, the functions on the left are ''defined'' by the equation on the right; these are not proved identities. The principal inverses are usually defined as:
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Function !! Definition !! Value Field
| |
| |-
| |
| | <math> \arcsin x = y \, </math>
| |
| | <math> \sin y = x \, </math>
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| | <math> -\frac{\pi}{2} \le y \le \frac{\pi}{2} \, </math>
| |
| |-
| |
| | <math> \arccos x = y \, </math>
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| | <math> \cos y = x \, </math>
| |
| | <math> 0 \le y \le \pi \, </math>
| |
| |-
| |
| | <math> \arctan x = y \, </math>
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| | <math> \tan y = x \, </math>
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| | <math> -\frac{\pi}{2} < y < \frac{\pi}{2} \, </math>
| |
| |-
| |
| | <math> \arccsc x = y \, </math>
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| | <math> \csc y = x \, </math>
| |
| | <math> -\frac{\pi}{2} \le y \le \frac{\pi}{2}, y \ne 0 \, </math>
| |
| |-
| |
| | <math> \arcsec x = y \, </math>
| |
| | <math> \sec y = x \, </math>
| |
| | <math> 0 \le y \le \pi, y \ne \frac{\pi}{2} \, </math>
| |
| |-
| |
| | <math> \arccot x = y \, </math>
| |
| | <math> \cot y = x \, </math>
| |
| | <math> 0 < y < \pi \, </math>
| |
| |}
| |
| | |
| The notations sin<sup>−1</sup> and cos<sup>−1</sup> are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "[[minute of arc|arcsecond]]".
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| | |
| Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,
| |
| :<math>
| |
| \arcsin z = z + \left( \frac {1} {2} \right) \frac {z^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{z^7} {7} + \cdots\,.</math>
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| These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:
| |
| | |
| :<math>
| |
| \arcsin z =
| |
| \int_0^z (1 - x^2)^{-1/2}\,dx, \quad |z| < 1.
| |
| </math>
| |
| | |
| Analogous formulas for the other functions can be found at [[Inverse trigonometric functions]]. Using the [[complex logarithm]], one can generalize all these functions to complex arguments:
| |
| | |
| :<math>
| |
| \arcsin z = -i \log \left( i z + \sqrt{1 - z^2} \right), \,
| |
| </math>
| |
| :<math>
| |
| \arccos z = -i \log \left( z + \sqrt{z^2 - 1}\right), \,
| |
| </math>
| |
| :<math>
| |
| \arctan z = \frac12i \log\left(\frac{1-iz}{1+iz}\right).
| |
| </math>
| |
| <!-- (note: these should probably be presented as definite integrals, removing the ambiguity of the constant) -->
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| | |
| ===Connection to the inner product===
| |
| In an [[inner product space]], the angle between two non-zero vectors is defined to be
| |
| :<math>\operatorname{angle}(x,y) = \arccos \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|}.</math>
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| | |
| ==Properties and applications==
| |
| {{Main|Uses of trigonometry}}
| |
| | |
| The trigonometric functions, as the name suggests, are of crucial importance in [[trigonometry]], mainly because of the following two results.
| |
| | |
| ===Law of sines===
| |
| The '''[[law of sines]]''' states that for an arbitrary [[triangle]] with sides ''a'', ''b'', and ''c'' and angles opposite those sides ''A'', ''B'' and ''C'':
| |
| | |
| :<math>\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{2\Delta}{abc},</math>
| |
| | |
| where <math>\Delta</math> is the area of the triangle,
| |
| or, equivalently,
| |
| | |
| :<math>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,</math>
| |
| | |
| where ''R'' is the triangle's [[circumscribed circle|circumradius]].
| |
| | |
| [[Image:Lissajous curve 5by4.svg|right|thumb|right|A [[Lissajous curve]], a figure formed with a trigonometry-based function.]]
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| | |
| It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in ''[[triangulation]]'', a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
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| | |
| ===Law of cosines===
| |
| The '''[[law of cosines]]''' (also known as the cosine formula or cosine rule) is an extension of the [[Pythagorean theorem]]:
| |
| | |
| :<math>c^2=a^2+b^2-2ab\cos C, \, </math>
| |
| or equivalently,
| |
| :<math>\cos C=\frac{a^2+b^2-c^2}{2ab}.</math>
| |
| | |
| In this formula the angle at ''C'' is opposite to the side ''c''. This theorem can be proven by dividing the triangle into two right ones and using the [[Pythagorean theorem]].
| |
| | |
| The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
| |
| | |
| ===Law of tangents===
| |
| {{main|Law of tangents}}
| |
| The following all form the [[law of tangents]]<ref>The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 529. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.</ref>
| |
| | |
| : <math>
| |
| \begin{align}
| |
| \frac{\tan \frac{A-B}{2 }}{\tan \frac{A+B}{2 } } & = \frac{a-b}{a+b} \\[6pt]
| |
| \frac{\tan \frac{A-C}{2 }}{\tan \frac{A+C}{2 } } & = \frac{a-c}{a+c} \\[6pt]
| |
| \frac{\tan \frac{B-C}{2 }}{\tan \frac{B+C}{2 } } & = \frac{b-c}{b+c}
| |
| \end{align}
| |
| </math>
| |
| | |
| The explanation of the formulae in words would be cumbersome, but the patterns of sums and differences; for the lengths and corresponding opposite angles, are apparent in the theorem.
| |
| | |
| ===Law of cotangents===
| |
| {{main|Law of cotangents}}
| |
| If
| |
| | |
| : <math> \zeta = \sqrt{\frac{1}{s} (s-a)(s-b)(s-c)} </math>
| |
| | |
| (the radius of the inscribed circle for the triangle) and
| |
| | |
| : <math> s = \frac{a+b+c}{2 } </math>
| |
| | |
| (the semi-perimeter for the triangle), then the following all form the [[law of cotangents]]<ref>The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.</ref>
| |
| | |
| : <math>\cot{ \frac{A}{2 }} = \frac{s-a}{\zeta }</math>
| |
| | |
| : <math>\cot{ \frac{B}{2 }} = \frac{s-b}{\zeta }</math>
| |
| | |
| : <math>\cot{ \frac{C}{2 }} = \frac{s-c}{\zeta }</math>
| |
| | |
| It follows that
| |
| | |
| : <math> \frac{\cot(A/2)}{s-a} = \frac{\cot(B/2)}{s-b} = \frac{\cot(C/2)}{s-c}. </math>
| |
| | |
| In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.
| |
| | |
| ===Other useful properties===
| |
| | |
| ====Sine and cosine of sums of angles====
| |
| {{See also|Angle addition formula}}
| |
| {{math|sin (α+β) {{=}} sin (α) cos (β) + cos (α) sin (β)}}
| |
| | |
| {{math|sin (α−β) {{=}} sin (α) cos (β) − cos (α) sin (β)}}
| |
| | |
| {{math|cos (α+β) {{=}} cos (α) cos (β) − sin (α) sin (β)}}
| |
| | |
| {{math|cos (α−β) {{=}} cos (α) cos (β) + sin (α) sin (β)}}
| |
| | |
| ===Periodic functions===
| |
| [[Image:Synthesis square.gif|thumb|340px|right|An animation of the [[additive synthesis]] of a [[square wave]] with an increasing number of harmonics]]
| |
| [[File:Sawtooth Fourier Animation.gif|thumb|280px|Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the [[Gibbs phenomenon]]]]
| |
| | |
| The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe [[simple harmonic motion]], which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of [[uniform circular motion]].
| |
| | |
| Trigonometric functions also prove to be useful in the study of general [[periodic function]]s. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light [[wave]]s.<ref name=Farlow>{{cite book |title=Partial differential equations for scientists and engineers |url=http://books.google.com/books?id=DLUYeSb49eAC&pg=PA82 |author=Stanley J Farlow |page=82 |isbn=0-486-67620-X |publisher=Courier Dover Publications |edition=Reprint of Wiley 1982 |year=1993}}</ref>
| |
| | |
| Under rather general conditions, a periodic function ''ƒ''(''x'') can be expressed as a sum of sine waves or cosine waves in a [[Fourier series]].<ref name=Folland>
| |
| | |
| See for example, {{cite book |author=Gerald B Folland |title=Fourier Analysis and its Applications |publisher=American Mathematical Society |edition=Reprint of Wadsworth & Brooks/Cole 1992 |url=http://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |pages = 77 ''ff'' |chapter=Convergence and completeness |year=2009 |isbn=0-8218-4790-2}}
| |
| | |
| </ref> Denoting the sine or cosine [[basis functions]] by ''φ<sub>k</sub>'', the expansion of the periodic function ''ƒ''(''t'') takes the form:
| |
| | |
| :<math>f(t) = \sum _{k=1}^\infty c_k \varphi_k(t). </math>
| |
| | |
| For example, the [[square wave]] can be written as the [[Fourier series]]
| |
| | |
| :<math> f_\text{square}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin \left ( (2k-1)t \right ) \over 2k-1}.</math>
| |
| | |
| In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a [[sawtooth wave]] are shown underneath.
| |
| | |
| ==History==
| |
| {{Main|History of trigonometric functions}}
| |
| While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The [[Chord (geometry)|chord]] function was discovered by [[Hipparchus]] of [[İznik|Nicaea]] (180–125 BC) and [[Ptolemy]] of [[Egypt (Roman province)|Roman Egypt]] (90–165 AD).
| |
| | |
| The functions sine and cosine can be traced to the [[Jyā, koti-jyā and utkrama-jyā|''jyā'' and ''koti-jyā '']] functions used in [[Gupta period]] [[Indian astronomy]] (''[[Aryabhatiya]]'', ''[[Surya Siddhanta]]''), via translation from Sanskrit to Arabic and then from Arabic to Latin.<ref>Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7, p. 210.
| |
| </ref>
| |
| | |
| All six trigonometric functions in current use were known in [[Islamic mathematics]] by the 9th century, as was the [[law of sines]], used in [[solving triangles]].<ref>
| |
| {{cite journal|title=Islamic Astronomy|author=Owen Gingerich|publisher=Scientific American|year=1986|volume=254|page=74|url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm|accessdate=2010-37-13}}</ref>
| |
| [[al-Khwārizmī]] produced tables of sines, cosines and tangents.
| |
| They were studied by authors including [[Omar Khayyám]], [[Bhāskara II]], [[Nasir al-Din al-Tusi]], [[Jamshīd al-Kāshī]] (14th century), [[Ulugh Beg]] (14th century), [[Regiomontanus]] (1464), [[Georg Joachim Rheticus|Rheticus]], and Rheticus' student [[Valentinus Otho]]{{Citation needed|date=June 2008}}
| |
| | |
| [[Madhava of Sangamagrama]] (c. 1400) made early strides in the [[mathematical analysis|analysis]] of trigonometric functions in terms of [[series (mathematics)|infinite series]].<ref name=mact-biog>{{cite web
| |
| | publisher=School of Mathematics and Statistics University of St Andrews, Scotland
| |
| | title=Madhava of Sangamagrama
| |
| |author = J J O'Connor and E F Robertson
| |
| |url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html
| |
| | accessdate=2007-09-08
| |
| }}
| |
| </ref>
| |
| | |
| The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician [[Albert Girard]].
| |
| | |
| In a paper published in 1682, [[Gottfried Leibniz|Leibniz]] proved that sin ''x'' is not an [[algebraic function]] of ''x''.<ref>{{cite book|title=Elements of the History of Mathematics|author=Nicolás Bourbaki|publisher=Springer|year=1994}}</ref>
| |
| | |
| [[Leonhard Euler]]'s ''Introductio in analysin infinitorum'' (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "[[Euler's formula]]", as well as the near-modern abbreviations ''sin., cos., tang., cot., sec.,'' and ''cosec.''<ref name=boyer>See Boyer (1991).</ref>
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| | |
| A few functions were common historically, but are now seldom used, such as the [[chord (geometry)|chord]] (crd(''θ'') = 2 sin(''θ''/2)), the [[versine]] (versin(''θ'') = 1 − cos(''θ'') = 2 sin<sup>2</sup>(''θ''/2)) (which appeared in the earliest tables<ref name=boyer/>), the [[Versine|haversine]] (haversin(''θ'') = versin(''θ'') / 2 = sin<sup>2</sup>(''θ''/2)), the [[exsecant]] (exsec(''θ'') = sec(''θ'') − 1) and the [[exsecant|excosecant]] (excsc(''θ'') = exsec(π/2 − ''θ'') = csc(''θ'') − 1). Many more relations between these functions are listed in the article about [[List of trigonometric identities|trigonometric identities]].
| |
| | |
| [[Etymology|Etymologically]], the word ''sine'' derives from the [[Sanskrit]] word for half the chord, ''jya-ardha'', abbreviated to ''jiva''. This was [[transliteration|transliterated]] in [[Arabic language|Arabic]] as ''jiba'', written ''jb'', vowels not being written in Arabic. Next, this transliteration was mis-translated in the 12th century into [[Latin]] as ''[[wikt:sinus|sinus]]'', under the mistaken impression that ''jb'' stood for the word ''jaib'', which means "bosom" or "bay" or "fold" in Arabic, as does ''sinus'' in Latin.<ref>See Maor (1998), chapter 3, regarding the etymology.</ref> Finally, English usage converted the Latin word ''sinus'' to ''sine''.<ref>{{cite web|url=http://www.clarku.edu/~djoyce/trig/|title=Clark University}}</ref> The word ''tangent'' comes from Latin ''tangens'' meaning "touching", since the line ''touches'' the circle of unit radius, whereas ''secant'' stems from Latin ''secans'' — "cutting" — since the line ''cuts'' the circle.
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| | |
| ==See also==
| |
| <div class="references" style="-moz-column-count:2; column-count:2;">
| |
| * [[All Students Take Calculus]] — a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane
| |
| * [[Aryabhata's sine table]]
| |
| * [[Bhaskara I's sine approximation formula]]
| |
| * [[Euler's formula]]
| |
| * [[Gauss's continued fraction]] — a [[continued fraction]] definition for the tangent function
| |
| * [[Generalized trigonometry]]
| |
| * [[Generating trigonometric tables]]
| |
| * [[Hyperbolic function]]
| |
| * [[List of periodic functions]]
| |
| * [[List of trigonometric identities]]
| |
| * [[Madhava series]]
| |
| * [[Madhava's sine table]]
| |
| * [[Polar sine]] — a generalization to vertex angles
| |
| * [[Proofs of trigonometric identities]]
| |
| * [[Table of Newtonian series]]
| |
| * [[Unit vector]] (explains direction cosines)
| |
| </div>
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| ==References==
| |
| {{refbegin}}
| |
| *Abramowitz, Milton and Irene A. Stegun, ''[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]'', Dover, New York. (1964). ISBN 0-486-61272-4.
| |
| *[[Lars Ahlfors]], ''Complex Analysis: an introduction to the theory of analytic functions of one complex variable'', second edition, [[McGraw-Hill Book Company]], New York, 1966.
| |
| *[[Carl Benjamin Boyer|Boyer, Carl B.]], ''A History of Mathematics'', John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
| |
| * Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transaction on Mathematical Software (1991).
| |
| *Joseph, George G., ''The Crest of the Peacock: Non-European Roots of Mathematics'', 2nd ed. [[Penguin Books]], London. (2000). ISBN 0-691-00659-8.
| |
| * Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," ''IEEE Trans. Computers'' '''45''' (3), 328–339 (1996).
| |
| *Maor, Eli, ''[http://www.pupress.princeton.edu/books/maor/ Trigonometric Delights]'', Princeton Univ. Press. (1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.
| |
| *Needham, Tristan, [http://www.usfca.edu/vca/PDF/vca-preface.pdf "Preface"]" to ''[http://www.usfca.edu/vca/ Visual Complex Analysis]''. Oxford University Press, (1999). ISBN 0-19-853446-9.
| |
| *O'Connor, J.J., and E.F. Robertson, [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html "Trigonometric functions"], ''[[MacTutor History of Mathematics archive]]''. (1996).
| |
| *O'Connor, J.J., and E.F. Robertson, [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html "Madhava of Sangamagramma"], ''[[MacTutor History of Mathematics archive]]''. (2000).
| |
| *Pearce, Ian G., [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_3.html "Madhava of Sangamagramma"], ''[[MacTutor History of Mathematics archive]]''. (2002).
| |
| *Weisstein, Eric W., [http://mathworld.wolfram.com/Tangent.html "Tangent"] from ''[[MathWorld]]'', accessed 21 January 2006.
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| {{Wikibooks|Trigonometry}}
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| * {{springer|title=Trigonometric functions|id=p/t094210}}
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| *[http://www.visionlearning.com/library/module_viewer.php?mid=131&l=&c3= Visionlearning Module on Wave Mathematics]
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| *[http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions
| |
| *[http://www.clarku.edu/~djoyce/trig/ Dave's draggable diagram.] (Requires java browser plugin)
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| {{DEFAULTSORT:Trigonometric Functions}}
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| [[Category:Trigonometry]]
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| [[Category:Elementary special functions]]
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| [[Category:Analytic functions]]
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| {{Link GA|pl}}
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| {{Link GA|zh}}
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| {{Link FA|ca}}
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| {{Link FA|km}}
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