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| [[File:Unit circle angles color.svg|300px|thumb|right|Cosines and sines around the [[unit circle]]]]
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| {{Trigonometry}}
| |
| In [[mathematics]], '''trigonometric identities''' are equalities that involve [[trigonometric functions]] and are true for every single value of the occurring [[Variable (mathematics)|variables]]. Geometrically, these are [[identity (mathematics)|identities]] involving certain functions of one or more [[angle]]s. They are distinct from [[Trigonometry#Triangle identities|triangle identities]], which are identities involving both angles and side lengths of a [[triangle]]. Only the former are covered in this article.
| |
| | |
| These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the [[integral|integration]] of non-trigonometric functions: a common technique involves first using the [[Trigonometric substitution|substitution rule with a trigonometric function]], and then simplifying the resulting integral with a trigonometric identity.
| |
| | |
| ==Notation==
| |
| ===Angles===
| |
| This article uses [[Greek letters]] such as [[Alpha (letter)|alpha]] (''α''), [[Beta (letter)|beta]] (''β''), [[gamma]] (''γ''), and [[theta]] (''θ'') to represent [[angle]]s. Several different [[Angle#Units|units of angle measure]] are widely used, including [[Degree (angle)|degrees]], [[radian]]s, and [[grad (angle)|grads]]: | |
| : 1 full circle = 360 degrees = 2<math>\pi</math> radians = 400 grads.
| |
| The following table shows the conversions for some common angles: | |
| {|class="wikitable" style="background-color: #FFFFFF; text-align: center;"
| |
| |-
| |
| ! [[Degree (angle)|Degree]]s
| |
| | 30° || 60° || 120° || 150°
| |
| | 210° || 240° || 300° || 330°
| |
| |-
| |
| ! [[Radian]]s
| |
| | <math>\frac\pi6\!</math> || <math>\frac\pi3\!</math> || <math>\frac{2\pi}3\!</math> || <math>\frac{5\pi}6\!</math>
| |
| | <math>\frac{7\pi}6\!</math> || <math>\frac{4\pi}3\!</math> || <math>\frac{5\pi}3\!</math> || <math>\frac{11\pi}6\!</math>
| |
| |-
| |
| ! [[Grad (angle)|Grad]]s
| |
| | 33⅓ grad || 66⅔ grad || 133⅓ grad || 166⅔ grad
| |
| | 233⅓ grad || 266⅔ grad || 333⅓ grad || 366⅔ grad
| |
| |-
| |
| |colspan="9"|
| |
| |-
| |
| ! [[Degree (angle)|Degree]]s
| |
| | 45° ||bgcolor="#F8F8FF"| 90° || 135° ||bgcolor="#F8F8FF"| 180°
| |
| | 225° || 270° || 315° ||bgcolor="#F8F8FF"| 360°
| |
| |-
| |
| ! [[Radian]]s
| |
| | <math>\frac\pi4\!</math> ||bgcolor="#F8F8FF"| <math>\frac\pi2\!</math> || <math>\frac{3\pi}4\!</math> ||bgcolor="#F8F8FF"| <math>\pi\!</math>
| |
| | <math>\frac{5\pi}4\!</math> || <math>\frac{3\pi}2\!</math> || <math>\frac{7\pi}4\!</math> ||bgcolor="#F8F8FF"| <math>2\pi\!</math>
| |
| |-
| |
| ! [[Grad (angle)|Grad]]s
| |
| | 50 grad ||bgcolor="#F8F8FF"| 100 grad || 150 grad ||bgcolor="#F8F8FF"| 200 grad
| |
| | 250 grad || 300 grad || 350 grad ||bgcolor="#F8F8FF"| 400 grad
| |
| |}
| |
| Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per [[Niven's theorem]] multiples of 30° are the only angles that are a [[Rational number|rational]] multiple of one degree and also have a rational sin/cos, which may account for their popularity in examples.<ref>Schaumberger, N. "A Classroom Theorem on Trigonometric Irrationalities." Two-Year College Math. J. 5, 73-76, 1974. also see Weisstein, Eric W. "Niven's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NivensTheorem.html</ref>
| |
| | |
| ===Trigonometric functions===
| |
| The primary trigonometric functions are the [[sine]] and [[cosine]] of an angle. These are sometimes abbreviated sin(''θ'') and cos(''θ''), respectively, where ''θ'' is the angle, but the parentheses around the angle are often omitted, e.g., sin ''θ'' and cos ''θ''.
| |
| | |
| The Sine of an angle is defined in the context of a [[Right Triangle]], as the ratio of the length of the side that is opposite to the angle, divided by the length of the longest side of the triangle (the [[Hypotenuse]] ).
| |
| | |
| The Cosine of an angle is also defined in the context of a [[Right Triangle]], as the ratio of the length of the side the angle is in, divided by the length of the longest side of the triangle (the [[Hypotenuse]] ).
| |
| | |
| The [[tangent function|tangent]] (tan) of an angle is the [[ratio]] of the sine to the cosine:
| |
| :<math>\tan\theta = \frac{\sin\theta}{\cos\theta}.</math>
| |
| Finally, the [[Trigonometric functions#Reciprocal functions|reciprocal functions]] secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:
| |
| :<math>\sec\theta = \frac{1}{\cos\theta},\quad\csc\theta = \frac{1}{\sin\theta},\quad\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}.</math>
| |
| These definitions are sometimes referred to as [[Proofs of trigonometric identities#Ratio identities|ratio identities]].
| |
| | |
| ==Inverse functions==
| |
| {{main|Inverse trigonometric functions}}
| |
| The inverse trigonometric functions are partial [[inverse function]]s for the trigonometric functions. For example, the inverse function for the sine, known as the '''inverse sine''' (sin<sup>−1</sup>) or '''arcsine''' (arcsin or asin), satisfies
| |
| :<math>\sin(\arcsin x) = x\quad\text{for} \quad |x| \leq 1 </math>
| |
| and
| |
| :<math>\arcsin(\sin x) = x\quad\text{for} \quad |x| \leq \pi/2. </math>
| |
| | |
| This article uses the notation below for inverse trigonometric functions:
| |
| {|class="wikitable" style="background-color: #FFFFFF; text-align: center;"
| |
| |-
| |
| ! Function
| |
| | sin
| |
| | cos
| |
| | tan
| |
| | sec
| |
| | csc
| |
| | cot
| |
| |-
| |
| ! Inverse
| |
| | arcsin
| |
| | arccos
| |
| | arctan
| |
| | arcsec
| |
| | arccsc
| |
| | arccot
| |
| |}
| |
| | |
| ==Pythagorean identity==
| |
| The basic relationship between the sine and the cosine is the [[Pythagorean trigonometric identity]]:
| |
| | |
| :<math>\cos^2\theta + \sin^2\theta = 1\!</math>
| |
| | |
| where {{math|cos<sup>2</sup> ''θ''}} means {{math|(cos(''θ''))<sup>2</sup>}} and {{math|sin<sup>2</sup> ''θ''}} means {{math|(sin(''θ''))<sup>2</sup>}}.
| |
| | |
| This can be viewed as a version of the [[Pythagorean theorem]], and follows from the equation {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 1}} for the [[unit circle]]. This equation can be solved for either the sine or the cosine:
| |
| | |
| :<math>\sin\theta = \pm \sqrt{1-\cos^2\theta} \quad \text{and} \quad \cos\theta = \pm \sqrt{1 - \sin^2\theta}. \, </math>
| |
| | |
| ===Related identities===
| |
| Dividing the Pythagorean identity by either {{math|cos<sup>2</sup> ''θ''}} or {{math|sin<sup>2</sup> ''θ''}} yields two other identities:
| |
| | |
| :<math>1 + \tan^2\theta = \sec^2\theta\quad\text{and}\quad 1 + \cot^2\theta = \csc^2\theta.\!</math>
| |
| | |
| Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other ([[up to]] a plus or minus sign):
| |
| | |
| <center>
| |
| {| class="wikitable" style="background-color:#FFFFFF;text-align:center"
| |
| |+ Each trigonometric function in terms of the other five.<ref>Abramowitz and Stegun, p. 73, 4.3.45</ref>
| |
| ! in terms of
| |
| ! scope="col" | <math> \sin \theta\!</math>
| |
| ! scope="col" | <math> \cos \theta\!</math>
| |
| ! scope="col" | <math> \tan \theta\!</math>
| |
| ! scope="col" | <math> \csc \theta\!</math>
| |
| ! scope="col" | <math> \sec \theta\!</math>
| |
| ! scope="col" | <math> \cot \theta\!</math>
| |
| |-
| |
| ! <math> \sin \theta =\!</math>
| |
| | <math> \sin \theta\ </math>
| |
| | <math>\pm\sqrt{1 - \cos^2 \theta}\! </math>
| |
| | <math>\pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}\! </math>
| |
| | <math> \frac{1}{\csc \theta}\! </math>
| |
| | <math>\pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}\! </math>
| |
| | <math>\pm\frac{1}{\sqrt{1 + \cot^2 \theta}}\! </math>
| |
| |-
| |
| ! <math> \cos \theta =\!</math>
| |
| | <math>\pm\sqrt{1 - \sin^2\theta}\! </math>
| |
| | <math> \cos \theta\! </math>
| |
| | <math>\pm\frac{1}{\sqrt{1 + \tan^2 \theta}}\! </math>
| |
| | <math>\pm\frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}\! </math>
| |
| | <math> \frac{1}{\sec \theta}\! </math>
| |
| | <math>\pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}\! </math>
| |
| |-
| |
| ! <math> \tan \theta =\!</math>
| |
| | <math>\pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\! </math>
| |
| | <math>\pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}\! </math>
| |
| | <math> \tan \theta\! </math>
| |
| | <math>\pm\frac{1}{\sqrt{\csc^2 \theta - 1}}\! </math>
| |
| | <math>\pm\sqrt{\sec^2 \theta - 1}\! </math>
| |
| | <math> \frac{1}{\cot \theta}\! </math>
| |
| |-
| |
| ! <math> \csc \theta =\!</math>
| |
| | <math> \frac{1}{\sin \theta}\! </math>
| |
| | <math>\pm\frac{1}{\sqrt{1 - \cos^2 \theta}}\! </math>
| |
| | <math>\pm\frac{\sqrt{1 + \tan^2 \theta}}{\tan \theta}\! </math>
| |
| | <math> \csc \theta\! </math>
| |
| | <math>\pm\frac{\sec \theta}{\sqrt{\sec^2 \theta - 1}}\! </math>
| |
| | <math>\pm\sqrt{1 + \cot^2 \theta}\! </math>
| |
| |-
| |
| ! <math> \sec \theta =\!</math>
| |
| | <math>\pm\frac{1}{\sqrt{1 - \sin^2 \theta}}\! </math><center>
| |
| | <math> \frac{1}{\cos \theta}\! </math>
| |
| | <math>\pm\sqrt{1 + \tan^2 \theta}\! </math>
| |
| | <math>\pm\frac{\csc \theta}{\sqrt{\csc^2 \theta - 1}}\! </math>
| |
| | <math> \sec \theta\! </math>
| |
| | <math>\pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}\! </math>
| |
| |-
| |
| ! <math> \cot \theta =\!</math>
| |
| | <math>\pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}\! </math>
| |
| | <math>\pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}\! </math>
| |
| | <math> \frac{1}{\tan \theta}\! </math>
| |
| | <math>\pm\sqrt{\csc^2 \theta - 1}\! </math>
| |
| | <math>\pm\frac{1}{\sqrt{\sec^2 \theta - 1}}\! </math>
| |
| | <math> \cot \theta\! </math>
| |
| |}</center>
| |
| | |
| ==Historic shorthands==
| |
| [[File:Circle-trig6.svg|300px|right|thumb|All of the trigonometric functions of an angle ''θ'' can be constructed geometrically in terms of a unit circle centered at ''O''. Many of these terms are no longer in common use.]]
| |
| The [[versine]], [[versine|coversine]], [[Versine|haversine]], and [[exsecant]] were used in navigation. For example the [[haversine formula]] was used to calculate the distance between two points on a sphere. They are rarely used today.
| |
| | |
| {|class="wikitable" style="background-color:#FFFFFF"
| |
| |-
| |
| ! Name(s)
| |
| ! Abbreviation(s)
| |
| ! Value<ref>Abramowitz and Stegun, p. 78, 4.3.147</ref>
| |
| |-
| |
| | versed sine, [[versine]]
| |
| || <math>\operatorname{versin}(\theta)</math><br /><math>\operatorname{vers}(\theta)</math><br /><math>\operatorname{ver}(\theta)</math>
| |
| || <math>1 - \cos (\theta)</math>
| |
| |-
| |
| | versed cosine, [[Versine|vercosine]]
| |
| || <math>\operatorname{vercosin}(\theta)</math>
| |
| || <math>1 + \cos (\theta)</math>
| |
| |-
| |
| | coversed sine, [[Versine|coversine]]
| |
| || <math>\operatorname{coversin}(\theta)</math><br /><math>\operatorname{cvs}(\theta)</math>
| |
| || <math>1 - \sin(\theta)</math>
| |
| |-
| |
| | coversed cosine, [[Versine|covercosine]]
| |
| || <math>\operatorname{covercosin}(\theta)</math>
| |
| || <math>1 + \sin(\theta)</math>
| |
| |-
| |
| | half versed sine, [[Versine|haversine]]
| |
| || <math>\operatorname{haversin}(\theta)</math>
| |
| || <math>\frac{1 - \cos (\theta)}{2}</math>
| |
| |-
| |
| | half versed cosine, [[Versine|havercosine]]
| |
| || <math>\operatorname{havercosin}(\theta)</math>
| |
| || <math>\frac{1 + \cos (\theta)}{2}</math>
| |
| |-
| |
| | half coversed sine, [[Versine|hacoversine]]<br />cohaversine
| |
| || <math>\operatorname{hacoversin}(\theta)</math>
| |
| || <math>\frac{1 - \sin (\theta)}{2}</math>
| |
| |-
| |
| | half coversed cosine, [[Versine|hacovercosine]]<br />cohavercosine
| |
| || <math>\operatorname{hacovercosin}(\theta)</math>
| |
| || <math>\frac{1 + \sin (\theta)}{2}</math>
| |
| |-
| |
| | exterior secant, [[exsecant]]
| |
| | <math>\operatorname{exsec}(\theta)</math>
| |
| | <math>\sec(\theta) - 1</math>
| |
| |-
| |
| | exterior cosecant, [[Exsecant|excosecant]]
| |
| | <math>\operatorname{excsc}(\theta)</math>
| |
| | <math>\csc(\theta) - 1</math>
| |
| |-
| |
| | [[chord (geometry)|chord]]
| |
| | <math>\operatorname{crd}(\theta)</math>
| |
| | <math>2\sin\frac{\theta}{2}</math>
| |
| |}
| |
| | |
| Ancient Indian mathematicians used [[Sanskrit]] terms [[Jyā, koti-jyā and utkrama-jyā]], based on the resemblance of the chord, arc, and radius to the shape of a bow and bowstring drawn back.
| |
| | |
| ==Symmetry, shifts, and periodicity==
| |
| By examining the unit circle, the following properties of the trigonometric functions can be established.
| |
| | |
| ===Symmetry===
| |
| When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities:
| |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| ! Reflected in <math>\theta=0 </math><ref>Abramowitz and Stegun, p. 72, 4.3.13–15</ref>
| |
| ! Reflected in <math>\theta= \pi/2</math><br/>(co-function identities)<ref>[http://jwbales.home.mindspring.com/precal/part5/part5.1.html The Elementary Identities<!-- Bot generated title -->]</ref>
| |
| ! Reflected in <math>\theta= \pi</math>
| |
| |-
| |
| |<math>
| |
| \begin{align}
| |
| \sin(-\theta) &= -\sin \theta \\
| |
| \cos(-\theta) &= +\cos \theta \\
| |
| \tan(-\theta) &= -\tan \theta \\
| |
| \csc(-\theta) &= -\csc \theta \\
| |
| \sec(-\theta) &= +\sec \theta \\
| |
| \cot(-\theta) &= -\cot \theta \\
| |
| \end{align}
| |
| </math>
| |
| |<math>
| |
| \begin{align}
| |
| \sin(\tfrac{\pi}{2} - \theta) &= +\cos \theta \\
| |
| \cos(\tfrac{\pi}{2} - \theta) &= +\sin \theta \\
| |
| \tan(\tfrac{\pi}{2} - \theta) &= +\cot \theta \\
| |
| \csc(\tfrac{\pi}{2} - \theta) &= +\sec \theta \\
| |
| \sec(\tfrac{\pi}{2} - \theta) &= +\csc \theta \\
| |
| \cot(\tfrac{\pi}{2} - \theta) &= +\tan \theta \\
| |
| \end{align}
| |
| </math>
| |
| |<math>
| |
| \begin{align}
| |
| \sin(\pi - \theta) &= +\sin \theta \\
| |
| \cos(\pi - \theta) &= -\cos \theta \\
| |
| \tan(\pi - \theta) &= -\tan \theta \\
| |
| \csc(\pi - \theta) &= +\csc \theta \\
| |
| \sec(\pi - \theta) &= -\sec \theta \\
| |
| \cot(\pi - \theta) &= -\cot \theta \\
| |
| \end{align}
| |
| </math>
| |
| |}
| |
| | |
| ===Shifts and periodicity===
| |
| By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express particular results more simply. Some examples of this are shown by shifting functions round by π/2, π and 2π radians. Because the periods of these functions are either π or 2π, there are cases where the new function is exactly the same as the old function without the shift.
| |
| | |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| !Shift by π/2
| |
| !Shift by π <br/> Period for tan and cot<ref>Abramowitz and Stegun, p. 72, 4.3.9</ref>
| |
| !Shift by 2π <br/> Period for sin, cos, csc and sec<ref>Abramowitz and Stegun, p. 72, 4.3.7–8</ref>
| |
| |-
| |
| |<math>
| |
| \begin{align}
| |
| \sin(\theta + \tfrac{\pi}{2}) &= +\cos \theta \\
| |
| \cos(\theta + \tfrac{\pi}{2}) &= -\sin \theta \\
| |
| \tan(\theta + \tfrac{\pi}{2}) &= -\cot \theta \\
| |
| \csc(\theta + \tfrac{\pi}{2}) &= +\sec \theta \\
| |
| \sec(\theta + \tfrac{\pi}{2}) &= -\csc \theta \\
| |
| \cot(\theta + \tfrac{\pi}{2}) &= -\tan \theta
| |
| \end{align}
| |
| </math>
| |
| |<math>
| |
| \begin{align}
| |
| \sin(\theta + \pi) &= -\sin \theta \\
| |
| \cos(\theta + \pi) &= -\cos \theta \\
| |
| \tan(\theta + \pi) &= +\tan \theta \\
| |
| \csc(\theta + \pi) &= -\csc \theta \\
| |
| \sec(\theta + \pi) &= -\sec \theta \\
| |
| \cot(\theta + \pi) &= +\cot \theta \\
| |
| \end{align}
| |
| </math>
| |
| |<math>
| |
| \begin{align}
| |
| \sin(\theta + 2\pi) &= +\sin \theta \\
| |
| \cos(\theta + 2\pi) &= +\cos \theta \\
| |
| \tan(\theta + 2\pi) &= +\tan \theta \\
| |
| \csc(\theta + 2\pi) &= +\csc \theta \\
| |
| \sec(\theta + 2\pi) &= +\sec \theta \\
| |
| \cot(\theta + 2\pi) &= +\cot \theta
| |
| \end{align}
| |
| </math>
| |
| |}
| |
| | |
| ==Angle sum and difference identities==
| |
| [[Image:AngleAdditionDiagram.svg|right|thumb|225px|Illustration of angle addition formulae for the sine and cosine. Emphasized segment is of unit length.]]
| |
| [[File:AngleAdditionDiagramTangent.svg|right|thumb|225px|Illustration of the angle addition formula for the tangent. Emphasized segments are of unit length.]]
| |
| | |
| {{see also|#Product-to-sum and sum-to-product identities}}
| |
| These are also known as the ''addition and subtraction theorems'' or ''formulae''.
| |
| They were originally established by the 10th century Persian mathematician [[Abū al-Wafā' Būzjānī]].
| |
| One method of proving these identities is to apply [[Euler's formula]]. The use of the symbols <math>\pm</math> and <math>\mp</math> is described in the article [[plus-minus sign]].
| |
| | |
| For the angle addition diagram for the sine and cosine, the line in bold with the 1 on it is of length 1. It is the hypotenuse of a right angle triangle with angle β which gives the sin β and cos β. The cos β line is the hypotenuse of a right angle triangle with angle α so it has sides sin α and cos α both multiplied by cos β. This is the same for the sin β line. The original line is also the hypotenuse of a right angle triangle with angle α+β, the opposite side is the sin(α+β) line up from the origin and the adjacent side is the cos(α+β) segment going horizontally from the top left.
| |
| | |
| Overall the diagram can be used to show the sine and cosine of sum identities
| |
| :<math>\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta</math>
| |
| :<math>\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta</math>
| |
| because the opposite sides of the rectangle are equal.
| |
| | |
| {|class="wikitable" style="background-color:#FFFFFF"
| |
| ! Sine
| |
| | align="center" | <math>\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \!</math><ref>Abramowitz and Stegun, p. 72, 4.3.16</ref><ref name="mathworld_addition">{{MathWorld|title=Trigonometric Addition Formulas|urlname=TrigonometricAdditionFormulas}}</ref>
| |
| |-
| |
| ! Cosine
| |
| | align="center" | <math>\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\,</math><ref name="mathworld_addition"/><ref>Abramowitz and Stegun, p. 72, 4.3.17</ref>
| |
| |-
| |
| ! Tangent
| |
| | align="center" | <math>\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}</math><ref name="mathworld_addition"/><ref>Abramowitz and Stegun, p. 72, 4.3.18</ref>
| |
| |-
| |
| ! Arcsine
| |
| | align="center" | <math>\arcsin\alpha \pm \arcsin\beta = \arcsin\left(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2}\right)</math><ref>Abramowitz and Stegun, p. 80, 4.4.42</ref>
| |
| |-
| |
| ! Arccosine
| |
| | align="center" | <math>\arccos\alpha \pm \arccos\beta = \arccos\left(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)}\right)</math><ref>Abramowitz and Stegun, p. 80, 4.4.43</ref>
| |
| |-
| |
| ! Arctangent
| |
| | align="center" | <math>\arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right)</math><ref>Abramowitz and Stegun, p. 80, 4.4.36</ref>
| |
| |}
| |
| | |
| === Matrix form ===
| |
| {{see also|matrix multiplication}}
| |
| | |
| The sum and difference formulae for sine and cosine can be written in [[matrix (mathematics)|matrix]] form as:
| |
| | |
| : <math>
| |
| \begin{align}
| |
| & {} \quad
| |
| \left(\begin{array}{rr}
| |
| \cos\alpha & -\sin\alpha \\
| |
| \sin\alpha & \cos\alpha
| |
| \end{array}\right)
| |
| \left(\begin{array}{rr}
| |
| \cos\beta & -\sin\beta \\
| |
| \sin\beta & \cos\beta
| |
| \end{array}\right) \\[12pt]
| |
| & = \left(\begin{array}{rr}
| |
| \cos\alpha\cos\beta - \sin\alpha\sin\beta & -\cos\alpha\sin\beta - \sin\alpha\cos\beta \\
| |
| \sin\alpha\cos\beta + \cos\alpha\sin\beta & -\sin\alpha\sin\beta + \cos\alpha\cos\beta
| |
| \end{array}\right) \\[12pt]
| |
| & = \left(\begin{array}{rr}
| |
| \cos(\alpha+\beta) & -\sin(\alpha+\beta) \\
| |
| \sin(\alpha+\beta) & \cos(\alpha+\beta)
| |
| \end{array}\right).
| |
| \end{align}
| |
| </math>
| |
| This shows that these matrices form a [[Group representation|representation]] of the rotation group in the plane (technically, the [[special orthogonal group]] ''SO''(2)), since the composition law is fulfilled: subsequent multiplications of a vector with these two matrices yields the same result as the rotation by the sum of the angles.
| |
| | |
| === Sines and cosines of sums of infinitely many terms ===
| |
| : <math> \sin\left(\sum_{i=1}^\infty \theta_i\right)
| |
| =\sum_{\text{odd}\ k \ge 1} (-1)^{(k-1)/2}
| |
| \sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\ \left|A\right| = k\end{smallmatrix}}
| |
| \left(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\right) </math>
| |
| | |
| : <math> \cos\left(\sum_{i=1}^\infty \theta_i\right)
| |
| =\sum_{\text{even}\ k \ge 0} ~ (-1)^{k/2} ~~
| |
| \sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\ \left|A\right| = k\end{smallmatrix}}
| |
| \left(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\right) </math>
| |
| | |
| In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and [[cofiniteness|cofinite]]ly many cosine factors.
| |
| | |
| If only finitely many of the terms θ<sub>''i''</sub> are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.
| |
| | |
| === Tangents of sums ===
| |
| Let ''e''<sub>''k''</sub> (for ''k'' = 0, 1, 2, 3, ...) be the ''k''th-degree [[elementary symmetric polynomial]] in the variables
| |
| | |
| : <math>x_i = \tan \theta_i\,</math>
| |
| | |
| for ''i'' = 0, 1, 2, 3, ..., i.e.,
| |
| | |
| : <math>
| |
| \begin{align}
| |
| e_0 & = 1 \\[6pt]
| |
| e_1 & = \sum_i x_i & & = \sum_i \tan\theta_i \\[6pt]
| |
| e_2 & = \sum_{i < j} x_i x_j & & = \sum_{i < j} \tan\theta_i \tan\theta_j \\[6pt]
| |
| e_3 & = \sum_{i < j < k} x_i x_j x_k & & = \sum_{i < j < k} \tan\theta_i \tan\theta_j \tan\theta_k \\
| |
| & {}\ \ \vdots & & {}\ \ \vdots
| |
| \end{align}
| |
| </math>
| |
| | |
| Then
| |
| | |
| : <math>\tan\left(\sum_i \theta_i\right) = \frac{e_1 - e_3 + e_5 -\cdots}{e_0 - e_2 + e_4 - \cdots}.\! </math>
| |
| | |
| The number of terms on the right side depends on the number of terms on the left side.
| |
| | |
| For example:
| |
| : <math> \begin{align}
| |
| \tan(\theta_1 + \theta_2) &
| |
| = \frac{ e_1 }{ e_0 - e_2 }
| |
| = \frac{ x_1 + x_2 }{ 1 \ - \ x_1 x_2 }
| |
| = \frac{ \tan\theta_1 + \tan\theta_2 }{ 1 \ - \ \tan\theta_1 \tan\theta_2 }
| |
| ,
| |
| \\[8pt]
| |
| \tan(\theta_1 + \theta_2 + \theta_3) &
| |
| = \frac{ e_1 - e_3 }{ e_0 - e_2 }
| |
| = \frac{ (x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3) }{ 1 \ - \ (x_1x_2 + x_1 x_3 + x_2 x_3) },
| |
| \\[8pt]
| |
| \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &
| |
| = \frac{ e_1 - e_3 }{ e_0 - e_2 + e_4 } \\[8pt] &
| |
| = \frac{ (x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4) }{ 1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4) },
| |
| \end{align}</math>
| |
| | |
| and so on. The case of only finitely many terms can be proved by [[mathematical induction]].<ref>{{cite conference |last=Bronstein |first=Manuel |title=Simplification of real elementary functions |pages=207–211 |doi=10.1145/74540.74566 |booktitle=Proceedings of the ACM-[[SIGSAM]] 1989 International Symposium on Symbolic and Algebraic Computation |editor= G. H. Gonnet (ed.) |conference=ISSAC'89 (Portland US-OR, 1989-07) |location=New York |publisher=[[Association for Computing Machinery|ACM]] |year=1989 |isbn=0-89791-325-6}}</ref>
| |
| | |
| === Secants and cosecants of sums ===
| |
| : <math>
| |
| \begin{align}
| |
| \sec\left(\sum_i \theta_i\right) & = \frac{\prod_i \sec\theta_i}{e_0 - e_2 + e_4 - \cdots} \\[8pt]
| |
| \csc\left(\sum_i \theta_i \right) & = \frac{\prod_i \sec\theta_i }{e_1 - e_3 + e_5 - \cdots}
| |
| \end{align}
| |
| </math>
| |
| | |
| where ''e''<sub>''k''</sub> is the ''k''th-degree [[elementary symmetric polynomial]] in the ''n'' variables ''x''<sub>''i''</sub> = tan ''θ''<sub>''i''</sub>, ''i'' = 1, ..., ''n'', and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left. The case of only finitely many terms can be proved by mathematical induction on the number of such terms. The convergence of the series in the denominators can be shown by writing the secant identity in the form
| |
| | |
| : <math> e_0 - e_2 + e_4 - \cdots = \frac{\prod_i \sec\theta_i}{\sec\left(\sum_i \theta_i\right)} </math>
| |
| | |
| and then observing that the left side converges if the right side converges, and similarly for the cosecant identity.
| |
| | |
| For example,
| |
| | |
| : <math>
| |
| \begin{align}
| |
| \sec(\alpha+\beta+\gamma) & = \frac{\sec\alpha \sec\beta \sec\gamma}{1 - \tan\alpha\tan\beta - \tan\alpha\tan\gamma - \tan\beta\tan\gamma } \\[8pt]
| |
| \csc(\alpha+\beta+\gamma) & = \frac{\sec\alpha \sec\beta \sec\gamma}{\tan\alpha + \tan\beta + \tan\gamma - \tan\alpha\tan\beta\tan\gamma}.
| |
| \end{align}
| |
| </math>
| |
| | |
| ==Multiple-angle formulae==
| |
| {|class="wikitable" style="background-color: #FFFFFF;"
| |
| |-
| |
| ! ''T<sub>n</sub>'' is the ''n''th [[Chebyshev polynomials|Chebyshev polynomial]]
| |
| | <math>\cos n\theta =T_n (\cos \theta )\,</math> <ref name = "mathworld_multiple_angle"/>
| |
| |-
| |
| ! ''S''<sub>''n''</sub> is the ''n''th [[spread polynomials|spread polynomial]]
| |
| | <math>\sin^2 n\theta = S_n (\sin^2\theta)\,</math>
| |
| |-
| |
| ! [[de Moivre's formula]], <math>i</math> is the [[imaginary unit]]
| |
| | <math>\cos n\theta +i\sin n\theta=(\cos(\theta)+i\sin(\theta))^n \,</math> <ref>Abramowitz and Stegun, p. 74, 4.3.48</ref>
| |
| |}
| |
| | |
| ===Double-angle, triple-angle, and half-angle formulae===
| |
| <!-- [[Double-angle formula]], [[Double-angle formula]], [[Triple-angle formula]], [[Triple-angle formula]], [[Half-angle formula]], and [[Half-angle formulae]] redirect here -->
| |
| {{see also|Tangent half-angle formula}}
| |
| These can be shown by using either the sum and difference identities or the multiple-angle formulae.
| |
| | |
| {|class="wikitable" style="background-color:#FFFFFF;"
| |
| !colspan="4"| Double-angle formulae<ref>Abramowitz and Stegun, p. 72, 4.3.24–26</ref><ref name="mathworld_double_angle">{{MathWorld|title=Double-Angle Formulas|urlname=Double-AngleFormulas}}</ref>
| |
| |-
| |
| |style="vertical-align:top"|<math>\begin{align}
| |
| \sin 2\theta &= 2 \sin \theta \cos \theta \ \\ &= \frac{2 \tan \theta} {1 + \tan^2 \theta}
| |
| \end{align}</math>
| |
| |<math>\begin{align}
| |
| \cos 2\theta &= \cos^2 \theta - \sin^2 \theta \\ &= 2 \cos^2 \theta - 1 \\
| |
| &= 1 - 2 \sin^2 \theta \\ &= \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}
| |
| \end{align}</math>
| |
| |<math>\tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\!</math>
| |
| |<math>\cot 2\theta = \frac{\cot^2 \theta - 1}{2 \cot \theta}\!</math>
| |
| |-
| |
| !colspan="4"| Triple-angle formulae<ref name="mathworld_multiple_angle">{{MathWorld|title=Multiple-Angle Formulas|urlname=Multiple-AngleFormulas}}</ref><ref>Abramowitz and Stegun, p. 72, 4.3.27–28</ref>
| |
| |-
| |
| |<math>\begin{align}\sin 3\theta & = - \sin^3\theta + 3 \cos^2\theta \sin\theta\\
| |
| & = - 4\sin^3\theta + 3\sin\theta \end{align}</math>
| |
| |<math>\begin{align}\cos 3\theta & = \cos^3\theta - 3 \sin^2 \theta\cos \theta \\
| |
| & = 4 \cos^3\theta - 3 \cos\theta\end{align}</math>
| |
| |<math>\tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta}\!</math>
| |
| |<math>\cot 3\theta = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta}\!</math>
| |
| |-
| |
| !colspan="4"| Half-angle formulae<ref>Abramowitz and Stegun, p. 72, 4.3.20–22</ref><ref name="mathworld_half_angle">{{MathWorld|title=Half-Angle Formulas|urlname=Half-AngleFormulas}}</ref>
| |
| |-
| |
| |<math>\begin{align}&\sin \frac{\theta}{2} = \sgn \!\! \left( \!\! 2 \pi \! - \! \theta \! + \! 4 \pi \! \left\lfloor \! \frac{\theta}{4\pi} \! \right\rfloor \! \right) \!\! \sqrt{\frac{1 \! - \! \cos \theta}{2}} \\ \\
| |
| &\left(\mathrm{or}\,\,\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2}\right)\end{align}</math>
| |
| |<math>\begin{align}&\cos \frac{\theta}{2} = \sgn \!\! \left(\!\! \pi \! + \! \theta \! + \! 4 \pi \! \left\lfloor \! \frac{\pi \! - \! \theta}{4\pi} \! \right\rfloor \! \right) \!\! \sqrt{\frac{1 + \cos\theta}{2}} \\ \\
| |
| &\left(\mathrm{or}\,\,\cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}\right)\end{align}</math>
| |
| |<math>\begin{align} \tan \frac{\theta}{2} &= \csc \theta - \cot \theta \\ &= \pm\, \sqrt{1 - \cos \theta \over 1 + \cos \theta} \\[8pt] &= \frac{\sin \theta}{1 + \cos \theta} \\[8pt] &= \frac{1-\cos \theta}{\sin \theta} \\[10pt]
| |
| \tan\frac{\eta+\theta}{2} & = \frac{\sin\eta+\sin\theta}{\cos\eta+\cos\theta} \\[8pt]
| |
| \tan\left(\frac{\theta}{2} + \frac{\pi}{4}\right) & = \sec\theta + \tan\theta \\[8pt]
| |
| \sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} & = \frac{1 - \tan(\theta/2)}{1 + \tan(\theta/2)} \\[8pt]
| |
| \tan\tfrac{1}{2}\theta & = \frac{\tan\theta}{1 + \sqrt{1+\tan^2\theta}} \\ &\mbox{for}\quad \theta \in \left(-\tfrac{\pi}{2},\tfrac{\pi}{2} \right)
| |
| \end{align}</math>
| |
| |<math>\begin{align} \cot \frac{\theta}{2} &= \csc \theta + \cot \theta \\ &= \pm\, \sqrt{1 + \cos \theta \over 1 - \cos \theta} \\[8pt] &= \frac{\sin \theta}{1 - \cos \theta} \\[8pt] &= \frac{1 + \cos \theta}{\sin \theta} \end{align}</math>
| |
| |}
| |
| The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a [[compass and straightedge constructions|compass and straightedge construction]] of [[angle trisection]] to the algebraic problem of solving a [[cubic function|cubic equation]], which allows one to prove that this is in general impossible using the given tools, by [[field (mathematics)|field theory]].
| |
| | |
| A formula for computing the trigonometric identities for the third-angle exists, but it requires finding the zeroes of the [[cubic function|cubic equation]] <math>x^3 - \frac{3x+d}{4}=0</math>, where ''x'' is the value of the sine function at some angle and ''d'' is the known value of the sine function at the triple angle. However, the [[discriminant]] of this equation is negative, so this equation has three real roots (of which only one is the solution within the correct third-circle) but none of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the [[cube root]]s, (which may be expressed in terms of real-only functions only if using hyperbolic functions).
| |
| | |
| === Sine, cosine, and tangent of multiple angles ===
| |
| For specific multiples, these follow from the angle addition formulas, while the general formula was given by 16th century French mathematician [[Franciscus Vieta|Vieta]].
| |
| | |
| :<math>\sin n\theta = \sum_{k=0}^n \binom{n}{k} \cos^k \theta\,\sin^{n-k} \theta\,\sin\left(\frac{1}{2}(n-k)\pi\right)</math>
| |
| | |
| :<math>\cos n\theta = \sum_{k=0}^n \binom{n}{k} \cos^k \theta\,\sin^{n-k} \theta\,\cos\left(\frac{1}{2}(n-k)\pi\right)</math>
| |
| | |
| In each of these two equations, the first parenthesized term is a [[binomial coefficient]], and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed. tan ''nθ'' can be written in terms of tan ''θ'' using the recurrence relation:
| |
| | |
| :<math>\tan\,(n{+}1)\theta = \frac{\tan n\theta + \tan \theta}{1 - \tan n\theta\,\tan \theta}.</math>
| |
| | |
| cot ''nθ'' can be written in terms of cot ''θ'' using the recurrence relation:
| |
| | |
| :<math>\cot\,(n{+}1)\theta = \frac{\cot n\theta\,\cot \theta - 1}{\cot n\theta + \cot \theta}.</math>
| |
| | |
| === Chebyshev method ===
| |
| The [[Pafnuty Chebyshev|Chebyshev]] method is a recursive algorithm for finding the ''n''<sup>th</sup> multiple angle formula knowing the (''n'' − 1)<sup>th</sup> and (''n'' − 2)<sup>th</sup> formulae.<ref>Ken Ward's Mathematics Pages, http://www.trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm</ref>
| |
| | |
| The cosine for ''nx'' can be computed from the cosine of (''n'' − 1)''x'' and (''n'' − 2)''x'' as follows:
| |
| | |
| :<math>\cos nx = 2 \cdot \cos x \cdot \cos ((n-1) x) - \cos ((n-2) x) \,</math>
| |
| | |
| Similarly sin(''nx'') can be computed from the sines of (''n'' − 1)''x'' and (''n'' − 2)''x''
| |
| | |
| :<math>\sin nx = 2 \cdot \cos x \cdot \sin ((n-1) x) - \sin ((n-2) x) \, </math>
| |
| | |
| For the tangent, we have:
| |
| | |
| :<math>\tan nx = \frac{H + K \tan x}{K- H \tan x} \, </math>
| |
| | |
| where ''H''/''K'' = tan(''n'' − 1)''x''.
| |
| | |
| === Tangent of an average ===
| |
| :<math> \tan\left( \frac{\alpha+\beta}{2} \right)
| |
| = \frac{\sin\alpha + \sin\beta}{\cos\alpha + \cos\beta}
| |
| = -\,\frac{\cos\alpha - \cos\beta}{\sin\alpha - \sin\beta}</math>
| |
| | |
| Setting either ''α'' or ''β'' to 0 gives the usual tangent half-angle formulæ.
| |
| | |
| === Viète's infinite product ===
| |
| : <math> \cos\left({\theta \over 2}\right) \cdot \cos\left({\theta \over 4}\right)
| |
| \cdot \cos\left({\theta \over 8}\right)\cdots = \prod_{n=1}^\infty \cos\left({\theta \over 2^n}\right)
| |
| = {\sin(\theta)\over \theta} = \operatorname{sinc}\,\theta. </math>
| |
| <!-- \operatorname{sinc} is intended to say "sinc", not "sin" and not "sine". --->
| |
| <!-- \operatorname{sinc} is intended to say "sinc", not "sin" and not "sine". --->
| |
| <!-- \operatorname{sinc} is intended to say "sinc", not "sin" and not "sine". --->
| |
| <!-- \operatorname{sinc} is intended to say "sinc", not "sin" and not "sine". --->
| |
| <!-- \operatorname{sinc} is intended to say "sinc", not "sin" and not "sine". --->
| |
| | |
| == Power-reduction formula ==
| |
| Obtained by solving the second and third versions of the cosine double-angle formula.
| |
| | |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| !Sine
| |
| !Cosine
| |
| !Other
| |
| |-
| |
| |<math>\sin^2\theta = \frac{1 - \cos 2\theta}{2}\!</math>
| |
| |<math>\cos^2\theta = \frac{1 + \cos 2\theta}{2}\!</math>
| |
| |<math>\sin^2\theta \cos^2\theta = \frac{1 - \cos 4\theta}{8}\!</math>
| |
| |-
| |
| |<math>\sin^3\theta = \frac{3 \sin\theta - \sin 3\theta}{4}\!</math>
| |
| |<math>\cos^3\theta = \frac{3 \cos\theta + \cos 3\theta}{4}\!</math>
| |
| |<math>\sin^3\theta \cos^3\theta = \frac{3\sin 2\theta - \sin 6\theta}{32}\!</math>
| |
| |-
| |
| |<math>\sin^4\theta = \frac{3 - 4 \cos 2\theta + \cos 4\theta}{8}\!</math>
| |
| |<math>\cos^4\theta = \frac{3 + 4 \cos 2\theta + \cos 4\theta}{8}\!</math>
| |
| |<math>\sin^4\theta \cos^4\theta = \frac{3-4\cos 4\theta + \cos 8\theta}{128}\!</math>
| |
| |-
| |
| |<math>\sin^5\theta = \frac{10 \sin\theta - 5 \sin 3\theta + \sin 5\theta}{16}\!</math>
| |
| |<math>\cos^5\theta = \frac{10 \cos\theta + 5 \cos 3\theta + \cos 5\theta}{16}\!</math>
| |
| |<math>\sin^5\theta \cos^5\theta = \frac{10\sin 2\theta - 5\sin 6\theta + \sin 10\theta}{512}\!</math>
| |
| |}
| |
| | |
| and in general terms of powers of {{nowrap|sin ''θ''}} or {{nowrap|cos ''θ''}} the following is true, and can be deduced using [[De Moivre's formula]], [[Euler's formula]] and [[binomial theorem]].
| |
| | |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| !
| |
| !Cosine
| |
| !Sine
| |
| |-
| |
| !<math>\text{if }n\text{ is odd}</math>
| |
| |<math>\cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} \cos{((n-2k)\theta)}</math>
| |
| |<math>\sin^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} (-1)^{(\frac{n-1}{2}-k)} \binom{n}{k} \sin{((n-2k)\theta)}</math>
| |
| |-
| |
| !<math>\text{if }n\text{ is even}</math>
| |
| |<math>\cos^n\theta = \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} \binom{n}{k} \cos{((n-2k)\theta)}</math>
| |
| |<math>\sin^n\theta = \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} (-1)^{(\frac{n}{2}-k)} \binom{n}{k} \cos{((n-2k)\theta)}</math>
| |
| |-
| |
| |}
| |
| | |
| ==Product-to-sum and sum-to-product identities==<!-- [[Standing wave]] links to this section -->
| |
| The product-to-sum identities or [[Prosthaphaeresis|prosthaphaeresis formulas]] can be proven by expanding their right-hand sides using the [[Trigonometric identity#Angle sum and difference identities|angle addition theorems]]. See [[beat (acoustics)]] and [[phase detector]] for applications of the sum-to-product formulæ.
| |
| | |
| {|
| |
| |style="vertical-align:top"|
| |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| !Product-to-sum<ref>Abramowitz and Stegun, p. 72, 4.3.31–33</ref>
| |
| |-
| |
| | <math>\cos \theta \cos \varphi = {{\cos(\theta - \varphi) + \cos(\theta + \varphi)} \over 2}</math>
| |
| |-
| |
| | <math>\sin \theta \sin \varphi = {{\cos(\theta - \varphi) - \cos(\theta + \varphi)} \over 2}</math>
| |
| |-
| |
| | <math>\sin \theta \cos \varphi = {{\sin(\theta + \varphi) + \sin(\theta - \varphi)} \over 2}</math>
| |
| |-
| |
| | <math>\cos \theta \sin \varphi = {{\sin(\theta + \varphi) - \sin(\theta - \varphi)} \over 2}</math>
| |
| |-
| |
| | <math>\tan \theta \tan \varphi =\frac{\cos(\theta-\varphi)-\cos(\theta+\varphi)}{\cos(\theta-\varphi)+\cos(\theta+\varphi)}</math>
| |
| |-\
| |
| | <math>\begin{align} \prod_{k=1}^n \cos \theta_k & = \frac{1}{2^n}\sum_{e\in S} \cos(e_1\theta_1+\cdots+e_n\theta_n) \\[6pt]
| |
| & \text{where }S=\{1,-1\}^n
| |
| \end{align}
| |
| </math>
| |
| |}
| |
| |
| |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| !Sum-to-product<ref>Abramowitz and Stegun, p. 72, 4.3.34–39</ref>
| |
| |-
| |
| |<math>\sin \theta \pm \sin \varphi = 2 \sin\left( \frac{\theta \pm \varphi}{2} \right) \cos\left( \frac{\theta \mp \varphi}{2} \right)</math>
| |
| |-
| |
| |<math>\cos \theta + \cos \varphi = 2 \cos\left( \frac{\theta + \varphi} {2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)</math>
| |
| |-
| |
| |<math>\cos \theta - \cos \varphi = -2\sin\left( {\theta + \varphi \over 2}\right) \sin\left({\theta - \varphi \over 2}\right)</math>
| |
| |}
| |
| |}
| |
| | |
| ===Other related identities===
| |
| :*<math>\text{If }x + y + z = \pi = \text{half circle,}\, </math>
| |
| | |
| :::<math>\text{then }\sin(2x) + \sin(2y) + \sin(2z) = 4\sin(x)\sin(y)\sin(z).\,</math>
| |
| | |
| :*(Triple tangent identity) <math>\text{If }x + y + z = \pi = \text{half circle,}\, </math>
| |
| | |
| :::<math>\text{then }\tan(x) + \tan(y) + \tan(z) = \tan(x)\tan(y)\tan(z).\,</math>
| |
| | |
| :::In particular, the formula holds when ''x'', ''y'', and ''z'' are the three angles of any triangle.
| |
| | |
| :::(If any of ''x'', ''y'', ''z'' is a right angle, one should take both sides to be ∞. This is neither +∞ nor −∞; for present purposes it makes sense to add just one point at infinity to the [[real number|real line]], that is approached by tan(θ) as tan(θ) either increases through positive values or decreases through negative values. This is a [[Alexandroff extension|one-point compactification]] of the real line.)
| |
| | |
| :*(Triple cotangent identity) <math>\text{If }x + y + z = \tfrac{\pi}{2} = \text{quarter circle,}\, </math>
| |
| | |
| :::<math>\text{then }\cot(x) + \cot(y) + \cot(z) = \cot(x)\cot(y)\cot(z).\,</math>
| |
| | |
| === Hermite's cotangent identity ===
| |
| {{main|Hermite's cotangent identity}}
| |
| | |
| [[Charles Hermite]] demonstrated the following identity.<ref>Warren P. Johnson, "Trigonometric Identities à la Hermite", ''[[American Mathematical Monthly]]'', volume 117, number 4, April 2010, pages 311–327</ref> Suppose ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> are [[complex number]]s, no two of which differ by an integer multiple of ''π''. Let
| |
| | |
| : <math> A_{n,k} = \prod_{\begin{smallmatrix} 1 \le j \le n \\ j \neq k \end{smallmatrix}} \cot(a_k - a_j) </math>
| |
| | |
| (in particular, ''A''<sub>1,1</sub>, being an [[empty product]], is 1). Then
| |
| | |
| : <math> \cot(z - a_1)\cdots\cot(z - a_n) = \cos\frac{n\pi}{2} + \sum_{k=1}^n A_{n,k} \cot(z - a_k).</math>
| |
| | |
| The simplest non-trivial example is the case ''n'' = 2:
| |
| | |
| : <math> \cot(z - a_1)\cot(z - a_2) = -1 + \cot(a_1 - a_2)\cot(z - a_1) + \cot(a_2 - a_1)\cot(z - a_2). </math>
| |
| | |
| === Ptolemy's theorem ===
| |
| :<math> \text{If }w + x + y + z = \pi = \text{half circle,} \, </math>
| |
| | |
| ::<math>\begin{align} \text{then }
| |
| & \sin(w + x)\sin(x + y) \\
| |
| &{} = \sin(x + y)\sin(y + z) \\
| |
| &{} = \sin(y + z)\sin(z + w) \\
| |
| &{} = \sin(z + w)\sin(w + x) = \sin(w)\sin(y) + \sin(x)\sin(z).
| |
| \end{align}</math>
| |
| | |
| (The first three equalities are trivial; the fourth is the substance of this identity.) Essentially this is [[Ptolemy's theorem]] adapted to the language of modern trigonometry.
| |
| | |
| ==Linear combinations==
| |
| For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different [[phase (waves)|phase shifts]] is also a sine wave with the same period or frequency, but a different phase shift. In the case of a non-zero linear combination of a sine and cosine wave<ref>Proof at http://pages.pacificcoast.net/~cazelais/252/lc-trig.pdf</ref> (which is just a sine wave with a phase shift of π/2), we have
| |
| | |
| :<math>a\sin x+b\cos x=\sqrt{a^2+b^2}\cdot\sin(x+\varphi)\,</math>
| |
| | |
| where
| |
| | |
| :<math>
| |
| \varphi = \begin{cases}\arcsin \left(\frac{b}{\sqrt{a^2+b^2}}\right)
| |
| & \text{if }a \ge 0, \\
| |
| \pi-\arcsin \left(\frac{b}{\sqrt{a^2+b^2}}\right) & \text{if }a < 0,
| |
| \end{cases}
| |
| </math>
| |
| | |
| or equivalently
| |
| | |
| :<math>
| |
| \varphi = \text{sgn}(b)\arccos \left(\tfrac{a}{\sqrt{a^2+b^2}}\right)
| |
| </math>
| |
| | |
| or even
| |
| | |
| :<math>
| |
| \varphi = \arctan \left(\frac{b}{a}\right) + \begin{cases}
| |
| 0 & \text{if }a \ge 0, \\
| |
| \pi & \text{if }a < 0,
| |
| \end{cases}
| |
| </math>
| |
| | |
| or using the [[atan2]] function
| |
| | |
| :<math> \varphi = \operatorname{atan2} \left( b, a \right).</math>
| |
| | |
| More generally, for an arbitrary phase shift, we have
| |
| | |
| :<math>a\sin x+b\sin(x+\alpha)= c \sin(x+\beta)\,</math>
| |
| | |
| where
| |
| <!--THIS FORMULA HAS A PLUS SIGN. TRY a = b = 1 AND phase alpha = 0 giving amplitude 2 -->
| |
| :<math>c = \sqrt{a^2 + b^2 + 2ab\cos \alpha},\,</math>
| |
| | |
| and
| |
| | |
| :<math>
| |
| \beta = \arctan \left(\frac{b\sin \alpha}{a + b\cos \alpha}\right) + \begin{cases}
| |
| 0 & \text{if } a + b\cos \alpha \ge 0, \\
| |
| \pi & \text{if } a + b\cos \alpha < 0.
| |
| \end{cases}
| |
| </math>
| |
| | |
| The general case reads{{citation needed|date=October 2012}}
| |
| :<math>\sum_i a_i \sin(x+\delta_i)= a \sin(x+\delta),</math>
| |
| where
| |
| :<math>a^2=\sum_{i,j}a_i a_j \cos(\delta_i-\delta_j)</math>
| |
| and
| |
| :<math>\tan \delta=\frac{\sum_i a_i \sin\delta_i}{\sum_i a_i \cos\delta_i}.</math>
| |
| | |
| See also [[Phasor_(sine_waves)#Addition|Phasor addition]].
| |
| | |
| ==Lagrange's trigonometric identities==
| |
| These identities, named after [[Joseph Louis Lagrange]], are:<ref name=Muniz>
| |
| {{cite journal |author=Eddie Ortiz Muñiz |date=February 1953 |volume=21 |number=2 |title=A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities |journal=American Journal of Physics |page=140|doi=10.1119/1.1933371}}
| |
| </ref><ref>
| |
| {{cite book |title=Handbook of Mathematical Formulas and Integrals |edition=4th |author=Alan Jeffrey and Hui-hui Dai |chapter=Section 2.4.1.6 |isbn=978-0-12-374288-9 |year=2008 |publisher=Academic Press}}
| |
| </ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \sum_{n=1}^N \sin n\theta & = \frac{1}{2}\cot\frac{\theta}{2}-\frac{\cos(N+\frac{1}{2})\theta}{2\sin\frac{1}{2}\theta}\\
| |
| \sum_{n=1}^N \cos n\theta & = -\frac{1}{2}+\frac{\sin(N+\frac{1}{2})\theta}{2\sin\frac{1}{2}\theta}
| |
| \end{align}
| |
| </math>
| |
| | |
| A related function is the following function of ''x'', called the [[Dirichlet kernel]].
| |
| | |
| : <math>1+2\cos(x) + 2\cos(2x) + 2\cos(3x) + \cdots + 2\cos(nx)
| |
| = \frac{\sin\left(\left(n +\frac{1}{2}\right)x\right)}{\sin(x/2)}.</math>
| |
| | |
| ==Other sums of trigonometric functions==
| |
| Sum of sines and cosines with arguments in arithmetic progression:<ref>Michael P. Knapp, [http://evergreen.loyola.edu/mpknapp/www/papers/knapp-sv.pdf Sines and Cosines of Angles in Arithmetic Progression]</ref> if <math>\alpha\ne0</math>, then
| |
| | |
| :<math>
| |
| \begin{align}
| |
| & \sin{\varphi} + \sin{(\varphi + \alpha)} + \sin{(\varphi + 2\alpha)} + \cdots {} \\[8pt]
| |
| & {} \qquad\qquad \cdots + \sin{(\varphi + n\alpha)} = \frac{\sin{\left(\frac{(n+1) \alpha}{2}\right)} \cdot \sin{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}} \quad\hbox{and}\\[10pt]
| |
| & \cos{\varphi} + \cos{(\varphi + \alpha)} + \cos{(\varphi + 2\alpha)} + \cdots {} \\[8pt]
| |
| & {} \qquad\qquad \cdots + \cos{(\varphi + n\alpha)} = \frac{\sin{\left(\frac{(n+1) \alpha}{2}\right)} \cdot \cos{(\varphi + \frac{n \alpha}{2})}}{\sin{\frac{\alpha}{2}}}.
| |
| \end{align}
| |
| </math>
| |
| | |
| For any ''a'' and ''b'':
| |
| : <math>a \cos(x) + b \sin(x) = \sqrt{ a^2 + b^2 } \cos(x - \operatorname{atan2}\,(b,a)) \;</math>
| |
| where [[atan2]](''y'', ''x'') is the generalization of [[Inverse trigonometric functions#Two-argument variant of arctangent|arctan]](''y''/''x'') that covers the entire circular range.
| |
| | |
| :<math>\tan(x) + \sec(x) = \tan\left({x \over 2} + {\pi \over 4}\right).</math>
| |
| The above identity is sometimes convenient to know when thinking about the [[Gudermannian function]], which relates the [[Trigonometric functions|circular]] and [[hyperbolic function|hyperbolic]] trigonometric functions without resorting to [[complex number]]s.
| |
| | |
| If ''x'', ''y'', and ''z'' are the three angles of any triangle, i.e. if ''x'' + ''y'' + ''z'' = π, then
| |
| :<math>\cot(x)\cot(y) + \cot(y)\cot(z) + \cot(z)\cot(x) = 1.\,</math>
| |
| | |
| == Certain linear fractional transformations ==
| |
| If ''ƒ''(''x'') is given by the [[Möbius transformation|linear fractional transformation]]
| |
| | |
| : <math> f(x) = \frac{(\cos\alpha)x - \sin\alpha}{(\sin\alpha)x + \cos\alpha}, </math>
| |
| | |
| and similarly
| |
| | |
| : <math> g(x) = \frac{(\cos\beta)x - \sin\beta}{(\sin\beta)x + \cos\beta}, </math>
| |
| | |
| then
| |
| | |
| : <math> f(g(x)) = g(f(x))
| |
| = \frac{(\cos(\alpha+\beta))x - \sin(\alpha+\beta)}{(\sin(\alpha+\beta))x + \cos(\alpha+\beta)}. </math>
| |
| | |
| More tersely stated, if for all ''α'' we let ''ƒ''<sub>''α''</sub> be what we called ''ƒ'' above, then
| |
| | |
| : <math> f_\alpha \circ f_\beta = f_{\alpha+\beta}. \, </math>
| |
| | |
| If ''x'' is the slope of a line, then ''ƒ''(''x'') is the slope of its rotation through an angle of −''α''.
| |
| | |
| ==Inverse trigonometric functions==
| |
| :<math> \arcsin(x)+\arccos(x)=\pi/2\;</math>
| |
| | |
| :<math> \arctan(x)+\arccot(x)=\pi/2.\;</math>
| |
| | |
| :<math>\arctan(x)+\arctan(1/x)=\left\{\begin{matrix} \pi/2, & \mbox{if }x > 0 \\ -\pi/2, & \mbox{if }x < 0 \end{matrix}\right.</math>
| |
| | |
| ===Compositions of trig and inverse trig functions===
| |
| {|class="wikitable" style="background-color: #FFFFFF"
| |
| |-
| |
| |<math>\sin[\arccos(x)]=\sqrt{1-x^2} \,</math>
| |
| |<math>\tan[\arcsin (x)]=\frac{x}{\sqrt{1 - x^2}}</math>
| |
| |-
| |
| |<math>\sin[\arctan(x)]=\frac{x}{\sqrt{1+x^2}}</math>
| |
| |<math>\tan[\arccos (x)]=\frac{\sqrt{1 - x^2}}{x}</math>
| |
| |-
| |
| |<math>\cos[\arctan(x)]=\frac{1}{\sqrt{1+x^2}}</math>
| |
| |<math>\cot[\arcsin (x)]=\frac{\sqrt{1 - x^2}}{x}</math>
| |
| |-
| |
| |<math>\cos[\arcsin(x)]=\sqrt{1-x^2} \,</math>
| |
| |<math>\cot[\arccos (x)]=\frac{x}{\sqrt{1 - x^2}}</math>
| |
| |}
| |
| | |
| ==Relation to the complex exponential function==
| |
| :<math>e^{ix} = \cos(x) +
| |
| i\sin(x)\,</math> <ref>Abramowitz and Stegun, p. 74,
| |
| 4.3.47</ref> ([[Euler's formula]]),
| |
| | |
| :<math>e^{-ix} = \cos(-x) + i\sin(-x) = \cos(x) - i\sin(x)</math>
| |
| | |
| :<math>e^{i\pi} = -1</math> ([[Euler's identity]]),
| |
| | |
| : <math>\cos(x) = \frac{e^{ix} + e^{-ix}}{2} </math><ref>Abramowitz and Stegun, p. 71,
| |
| 4.3.2</ref>
| |
| | |
| : <math>\sin(x) = \frac{e^{ix} - e^{-ix}}{2i} </math><ref>Abramowitz and Stegun, p. 71,
| |
| 4.3.1</ref>
| |
| | |
| and hence the corollary:
| |
| | |
| : <math>\tan(x) = \frac{\sin(x)}{\cos(x)}= \frac{e^{ix} - e^{-ix}}{i({e^{ix} + e^{-ix}})}
| |
| </math>
| |
| | |
| where <math>i^2 = -1</math>.
| |
| | |
| ==Infinite product formulae==
| |
| For applications to [[special functions]], the following [[infinite product]] formulae for trigonometric functions are useful:<ref>Abramowitz and Stegun, p. 75, 4.3.89–90</ref><ref>Abramowitz and Stegun, p. 85, 4.5.68–69</ref>
| |
| {{col-start}}
| |
| {{col-2}}
| |
| : <math>\sin x = x \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2 n^2}\right)</math>
| |
| | |
| : <math>\sinh x = x \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2 n^2}\right)</math>
| |
| | |
| : <math>\frac{\sin x}{x} = \prod_{n = 1}^\infty\cos\left(\frac{x}{2^n}\right)</math>
| |
| {{col-2}}
| |
| : <math>\cos x = \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2(n - \frac{1}{2})^2}\right)</math>
| |
| | |
| : <math>\cosh x = \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2(n - \frac{1}{2})^2}\right)</math>
| |
| | |
| : <math>|\sin x| = \frac1{2}\prod_{n = 0}^\infty \sqrt[2^{n+1}]{\left|\tan\left(2^n x\right)\right|}</math>
| |
| {{col-end}}
| |
| | |
| ==Identities without variables==
| |
| The [[Morrie's law|curious identity]]
| |
| | |
| :<math>\cos 20^\circ\cdot\cos 40^\circ\cdot\cos 80^\circ=\frac{1}{8}</math>
| |
| is a special case of an identity that contains one variable:
| |
| | |
| :<math>\prod_{j=0}^{k-1}\cos(2^j x)=\frac{\sin(2^k x)}{2^k\sin(x)}.</math>
| |
| | |
| Similarly:
| |
| | |
| :<math>\sin 20^\circ\cdot\sin 40^\circ\cdot\sin 80^\circ=\frac{\sqrt{3}}{8}.</math>
| |
| | |
| The same cosine identity in radians is
| |
| | |
| :<math> \cos\frac{\pi}{9}\cos\frac{2\pi}{9}\cos\frac{4\pi}{9} = \frac{1}{8}, </math>
| |
| | |
| Similarly:
| |
| | |
| :<math>\tan 50^\circ\cdot\tan 60^\circ\cdot\tan 70^\circ=\tan 80^\circ.</math>
| |
| | |
| :<math>\tan 40^\circ\cdot\tan 30^\circ\cdot\tan 20^\circ=\tan 10^\circ.</math>
| |
| | |
| The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
| |
| | |
| :<math>\cos 24^\circ+\cos 48^\circ+\cos 96^\circ+\cos 168^\circ=\frac{1}{2}.</math>
| |
| | |
| Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
| |
| :<math>
| |
| \begin{align}
| |
| & \cos\left( \frac{2\pi}{21}\right)
| |
| + \cos\left(2\cdot\frac{2\pi}{21}\right)
| |
| + \cos\left(4\cdot\frac{2\pi}{21}\right) \\[10pt]
| |
| & {} \qquad {} + \cos\left( 5\cdot\frac{2\pi}{21}\right)
| |
| + \cos\left( 8\cdot\frac{2\pi}{21}\right)
| |
| + \cos\left(10\cdot\frac{2\pi}{21}\right)=\frac{1}{2}.
| |
| \end{align}
| |
| </math>
| |
| | |
| The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are [[Coprime|relatively prime]] to (or have no [[prime factor]]s in common with) 21. The last several examples are corollaries of a basic fact about the irreducible [[cyclotomic polynomial]]s: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the [[Möbius function]] evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
| |
| | |
| Many of those curious identities stem from more general facts like the following:<ref>[[Eric W. Weisstein|Weisstein, Eric W.]], "[http://mathworld.wolfram.com/Sine.html Sine]" from [[MathWorld]]</ref>
| |
| | |
| :<math> \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}</math>
| |
| | |
| and
| |
| | |
| :<math> \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}} </math>
| |
| | |
| Combining these gives us
| |
| | |
| :<math> \prod_{k=1}^{n-1} \tan\left(\frac{k\pi}{n}\right) = \frac{n}{\sin(\pi n/2)}</math>
| |
| | |
| If ''n'' is an odd number (''n'' = 2''m'' + 1) we can make use of the symmetries to get
| |
| | |
| :<math> \prod_{k=1}^{m} \tan\left(\frac{k\pi}{2m+1}\right) = \sqrt{2m+1}</math>
| |
| | |
| The transfer function of the [[Butterworth filter|Butterworth low pass filter]] can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:
| |
| | |
| :<math> \prod_{k=1}^{n} \sin\left(\frac{\left(2k-1\right)\pi}{4n}\right) = \prod_{k=1}^{n} \cos\left(\frac{\left(2k-1\right)\pi}{4n}\right) = \frac{\sqrt{2}}{2^{n}}</math>
| |
| | |
| === Computing π ===
| |
| An efficient way to [[pi|compute π]] is based on the following identity without variables, due to [[John Machin|Machin]]:
| |
| | |
| :<math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239}</math>
| |
| | |
| or, alternatively, by using an identity of [[Leonhard Euler]]:
| |
| | |
| :<math>\frac{\pi}{4} = 5 \arctan\frac{1}{7} + 2 \arctan\frac{3}{79}.</math>
| |
| | |
| === A useful mnemonic for certain values of sines and cosines ===
| |
| For certain simple angles, the sines and cosines take the form <math>\scriptstyle\sqrt{n}/2</math> for 0 ≤ ''n'' ≤ 4, which makes them easy to remember.
| |
| | |
| :<math>
| |
| \begin{matrix}
| |
| \sin 0 & = & \sin 0^\circ & = & \sqrt{0}/2 & = & \cos 90^\circ & = & \cos \left( \frac {\pi} {2} \right) \\ \\
| |
| \sin \left( \frac {\pi} {6} \right) & = & \sin 30^\circ & = & \sqrt{1}/2 & = & \cos 60^\circ & = & \cos \left( \frac {\pi} {3} \right) \\ \\
| |
| \sin \left( \frac {\pi} {4} \right) & = & \sin 45^\circ & = & \sqrt{2}/2 & = & \cos 45^\circ & = & \cos \left( \frac {\pi} {4} \right) \\ \\
| |
| \sin \left( \frac {\pi} {3} \right) & = & \sin 60^\circ & = & \sqrt{3}/2 & = & \cos 30^\circ & = & \cos \left( \frac {\pi} {6} \right)\\ \\
| |
| \sin \left( \frac {\pi} {2} \right) & = & \sin 90^\circ & = & \sqrt{4}/2 & = & \cos 0^\circ & = & \cos 0
| |
| \end{matrix}
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| </math>
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| ===Miscellany===
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| With the [[golden ratio]] φ:
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| :<math>\cos \left( \frac {\pi} {5} \right) = \cos 36^\circ={\sqrt{5}+1 \over 4} = \frac{\varphi }{2}
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| </math>
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| :<math>\sin \left( \frac {\pi} {10} \right) = \sin 18^\circ = {\sqrt{5}-1 \over 4} = {\varphi - 1 \over 2} = {1 \over 2\varphi}</math>
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| Also see [[exact trigonometric constants]].
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| === An identity of Euclid ===
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| [[Euclid]] showed in Book XIII, Proposition 10 of his ''[[Euclid's Elements|Elements]]'' that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
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| :<math>\sin^2(18^\circ)+\sin^2(30^\circ)=\sin^2(36^\circ). \, </math>
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| [[Ptolemy]] used this proposition to compute some angles in [[Ptolemy's table of chords|his table of chords]].
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| == Composition of trigonometric functions ==
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| This identity involves a trigonometric function of a trigonometric function:<ref>[[Abramowitz and Stegun|Milton Abramowitz and Irene Stegun, ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'']], [[Dover Publications]], New York, 1972, formula 9.1.42</ref>
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| : <math>\cos(t \sin(x)) = J_0(t) + 2 \sum_{k=1}^\infty J_{2k}(t) \cos(2kx) </math>
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| where ''J''<sub>0</sub> and ''J''<sub>2''k''</sub> are [[Bessel function]]s.
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| ==Calculus==
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| In [[calculus]] the relations stated below require angles to be measured in [[radian]]s; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, along with the definitions of [[arc length]] and [[Jordan measure|area]], their derivatives can be found by verifying two limits. The first is:
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| :<math>\lim_{x\rightarrow 0}\frac{\sin x}{x}=1,</math>
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| verified using the [[unit circle]] and [[squeeze theorem]]. The second limit is:
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| :<math>\lim_{x\rightarrow 0}\frac{1-\cos x }{x}=0,</math>
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| verified using the identity tan(''x''/2) = (1 − cos ''x'')/sin ''x''. Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that (sin ''x'')′ = cos ''x'' and (cos ''x'')′ = −sin ''x''. If the sine and cosine functions are defined by their [[Taylor series]], then the derivatives can be found by differentiating the power series term-by-term.
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| :<math>{\mathrm{d} \over \mathrm{d}x}\sin x = \cos x</math>
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| The rest of the trigonometric functions can be differentiated using the above identities and the rules of [[derivative|differentiation]]:<ref>Abramowitz and Stegun, p. 77, 4.3.105–110</ref><ref>Abramowitz and Stegun, p. 82, 4.4.52–57</ref><ref>{{cite book
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| | last =Finney
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| | first =Ross
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| | title =Calculus : Graphical, Numerical, Algebraic
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| | publisher =Prentice Hall
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| | year =2003
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| | location =Glenview, Illinois
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| | pages =159–161
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| | isbn =0-13-063131-0 }}
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| </ref>
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| :<math>
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| \begin{align}
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| {\mathrm{d} \over \mathrm{d}x} \sin x & = \cos x, & {\mathrm{d} \over \mathrm{d}x} \arcsin x & = {1 \over \sqrt{1 - x^2}} \\ \\
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| {\mathrm{d} \over \mathrm{d}x} \cos x & = -\sin x, & {\mathrm{d} \over \mathrm{d}x} \arccos x & = {-1 \over \sqrt{1 - x^2}} \\ \\
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| {\mathrm{d} \over \mathrm{d}x} \tan x & = \sec^2 x, & {\mathrm{d} \over \mathrm{d}x} \arctan x & = { 1 \over 1 + x^2} \\ \\
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| {\mathrm{d} \over \mathrm{d}x} \cot x & = -\csc^2 x, & {\mathrm{d} \over \mathrm{d}x} \arccot x & = {-1 \over 1 + x^2} \\ \\
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| {\mathrm{d} \over \mathrm{d}x} \sec x & = \tan x \sec x, & {\mathrm{d} \over \mathrm{d}x} \arcsec x & = { 1 \over |x|\sqrt{x^2 - 1}} \\ \\
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| {\mathrm{d} \over \mathrm{d}x} \csc x & = -\csc x \cot x, & {\mathrm{d} \over \mathrm{d}x} \arccsc x & = {-1 \over |x|\sqrt{x^2 - 1}}
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| \end{align}
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| </math>
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| The integral identities can be found in "[[list of integrals of trigonometric functions]]". Some generic forms are listed below.
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| :<math>\int \frac{\mathrm{d}u}{\sqrt{a^2-u^2}} =\sin^{-1}\left( \frac{u}{a} \right)+C</math>
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| :<math>\int \frac{\mathrm{d}u}{a^2+u^2} =\frac{1}{a}\tan ^{-1}\left( \frac{u}{a} \right)+C</math>
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| :<math>\int \frac{\mathrm{d}u}{u\sqrt{u^2-a^2}} =\frac{1}{a}\sec ^{-1}\left| \frac{u}{a} \right|+C</math>
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| ===Implications===
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| The fact that the differentiation of trigonometric functions (sine and cosine) results in [[linear combination]]s of the same two functions is of fundamental importance to many fields of mathematics, including [[differential equation]]s and [[Fourier transform]]s.
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| === Some differential equations satisfied by the sine ===
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| Let ''i'' = √−1 be the imaginary unit and let <math>\circ</math> denote composition of differential operators. Then for every '''odd''' positive integer ''n'',
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| : <math> \sum_{k=0}^n \binom{n}{k} \left(\frac{\mathrm{d}}{\mathrm{d}x}-\sin x\right) \circ \left(\frac{\mathrm{d}}{\mathrm{d}x} - \sin x + i\right)\circ\cdots\circ\left(\frac{\mathrm{d}}{\mathrm{d}x}-\sin x+(k-1)i\right) (\sin x)^{n-k} = 0. </math>
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| (When ''k'' = 0, then the number of differential operators being composed is 0, so the corresponding term in the sum above is just (sin ''x'')<sup>''n''</sup>.) This identity was discovered as a by-product of research in [[medical imaging]].<ref>Peter Kuchment and Sergey Lvin, "Identities for sin ''x'' that Came from Medical Imaging", ''[[American Mathematical Monthly]]'', volume 120, August–September, 2013, pages 609–621.</ref>
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| == Exponential definitions ==
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| {| class="wikitable" style="background-color:#FFFFFF"
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| !Function
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| !Inverse function<ref>Abramowitz and Stegun, p. 80, 4.4.26–31</ref>
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| |-
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| |<math>\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \,</math>
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| |<math>\arcsin x = -i \ln \left(ix + \sqrt{1 - x^2}\right) \,</math>
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| |-
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| |<math>\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \,</math>
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| |<math>\arccos x = i\,\ln\left(x-i\,\sqrt{1-x^2}\right) \,</math>
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| |-
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| |<math>\tan \theta = \frac{e^{i\theta} - e^{-i\theta}}{i(e^{i\theta} + e^{-i\theta})} \,</math>
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| |<math>\arctan x = \frac{i}{2} \ln \left(\frac{i + x}{i - x}\right) \,</math>
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| |-
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| |<math>\csc \theta = \frac{2i}{e^{i\theta} - e^{-i\theta}} \,</math>
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| |<math>\arccsc x = -i \ln \left(\tfrac{i}{x} + \sqrt{1 - \tfrac{1}{x^2}}\right) \,</math>
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| |-
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| |<math>\sec \theta = \frac{2}{e^{i\theta} + e^{-i\theta}} \,</math>
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| |<math>\arcsec x = -i \ln \left(\tfrac{1}{x} + \sqrt{1 - \tfrac{i}{x^2}}\right) \,</math>
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| |-
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| |<math>\cot \theta = \frac{i(e^{i\theta} + e^{-i\theta})}{e^{i\theta} - e^{-i\theta}} \,</math>
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| |<math>\arccot x = \frac{i}{2} \ln \left(\frac{x - i}{x + i}\right) \,</math>
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| |-
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| !
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| !
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| |-
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| |<math>\operatorname{cis} \, \theta = e^{i\theta} \,</math>
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| |<math>\operatorname{arccis} \, x = \frac{\ln x}{i} = -i \ln x = \operatorname{arg} \, x \,</math>
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| |}
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| ==Miscellaneous==<!--This section will hopefully be sorted back into the article, If I can work out a place for the stuff to go-->
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| ===Dirichlet kernel===
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| The '''[[Dirichlet kernel]]''' ''D<sub>n</sub>''(''x'') is the function occurring on both sides of the next identity:
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| :<math>1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots+2\cos(nx) = \frac{ \sin\left[\left(n+\frac{1}{2}\right)x\right\rbrack }{ \sin\left(\frac{x}{2}\right) }. </math>
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| The [[convolution]] of any [[integrable function]] of period 2π with the Dirichlet kernel coincides with the function's ''n''th-degree Fourier approximation. The same holds for any [[Measure (mathematics)|measure]] or [[distribution (mathematics)|generalized function]].
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| === Tangent half-angle substitution ===
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| {{main|Tangent half-angle substitution}}
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| If we set
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| :<math>t = \tan\left(\frac{x}{2}\right),</math>
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| then<ref>Abramowitz and Stegun, p. 72, 4.3.23</ref>
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| : <math>\sin(x) = \frac{2t}{1 + t^2}\text{ and }\cos(x) = \frac{1 - t^2}{1 + t^2}\text{ and }e^{i x} = \frac{1 + i t}{1 - i t}</math>
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| where e<sup>''ix''</sup> = cos(''x'') + ''i'' sin(''x''), sometimes abbreviated to cis(''x'').
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| When this substitution of ''t'' for tan(''x''/2) is used in [[calculus]], it follows that sin(''x'') is replaced by 2''t''/(1 + ''t''<sup>2</sup>), cos(''x'') is replaced by (1 − ''t''<sup>2</sup>)/(1 + ''t''<sup>2</sup>) and the differential ''dx'' is replaced by (2 ''dt'')/(1 + ''t''<sup>2</sup>). Thereby one converts rational functions of sin(''x'') and cos(''x'') to rational functions of ''t'' in order to find their antiderivatives.
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| ==See also==
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| <div style="-moz-column-count:2; column-count:2;">
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| *[[Table of derivatives#Derivatives of trigonometric functions|Derivatives of trigonometric functions]]
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| *[[Exact trigonometric constants]] (values of sine and cosine expressed in surds)
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| *[[Exsecant]]
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| *[[Half-side formula]]
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| *[[Hyperbolic function]]
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| *[[Law of cosines]]
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| *[[Law of sines]]
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| *[[Law of tangents]]
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| *[[List of integrals of trigonometric functions]]
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| *[[Mollweide's formula]]
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| *[[Proofs of trigonometric identities]]
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| *[[Prosthaphaeresis]]
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| *[[Pythagorean theorem]]
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| *[[Tangent half-angle formula]]
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| *[[Trigonometry]]
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| *[[Uses of trigonometry]]
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| *[[Versine|Versine and haversine]]
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| </div>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{Cite book | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61272-0 | year=1972 | postscript=<!--None-->}}
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| ==External links==
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| *[http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Values of Sin and Cos, expressed in surds, for integer multiples of 3° and of 5⅝°], and for the same angles [http://www.jdawiseman.com/papers/easymath/surds_csc_sec.html Csc and Sec] and [http://www.jdawiseman.com/papers/easymath/surds_tan.html Tan].
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| [[Category:Mathematical identities]]
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| [[Category:Trigonometry]]
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| [[Category:Mathematics-related lists|Trigonometric identities]]
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| {{Link FA|lmo}}
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