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{{Trigonometry}}
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{|class="wikitable" style="background-color:#FFFFFF; margin:1em 0 1em 1em" align="right" hspace="3"
! Function
! Derivative
|-
| <math>\sin(x)</math>
| <math>\cos(x)</math>
|-
|<math>\cos(x)</math>
|<math>-\sin(x)</math>
|-
|<math>\tan(x)</math>
|<math>\sec^2(x)</math>
|-
|<math>\cot(x)</math>
|<math>-\csc^2(x)</math>
|-
|<math>\sec(x)</math>
|<math>\sec(x)\tan(x)</math>
|-
|<math>\csc(x)</math>
|<math>-\csc(x)\cot(x)</math>
|-
|<math>\arcsin(x)</math>
|<math>\frac{1}{\sqrt{1-x^2}}</math>
|-
|<math>\arccos(x)</math>
|<math>\frac{-1}{\sqrt{1-x^2}}</math>
|-
|<math>\arctan(x)</math>
|<math>\frac{1}{x^2+1}</math>
|}
The '''differentiation of trigonometric functions''' is the mathematical process of finding the rate at which a [[trigonometric function]] changes with respect to a variable—the [[derivative]] of the trigonometric function. Commonplace trigonometric functions include sin(''x''), cos(''x'') and tan(''x''). For example, in differentiating ''f''(''x'') = sin(''x''), one is calculating a function ''f''&nbsp;′(''x'') which computes the rate of change of sin(''x'') at a particular point ''a''. The value of the rate of change at ''a'' is thus given by ''f''&nbsp;′(''a''). Knowledge of differentiation from first principles is required, along with competence in the use of [[trigonometric identities]] and limits. All functions involve the arbitrary variable ''x'', with all differentiation performed with respect to ''x''.


It turns out that once one knows the derivatives of sin(''x'') and cos(''x''), one can easily compute the derivatives of the other circular trigonometric functions because they can all be expressed in terms of sine or cosine; the [[quotient rule]] is then implemented to differentiate this expression. Proofs of the derivatives of sin(''x'') and cos(''x'') are given in the proofs section; the results are quoted in order to give proofs of the derivatives of the other circular trigonometric functions. Finding the derivatives of the [[inverse trigonometric functions]] involves using [[implicit differentiation]] and the derivatives of regular trigonometric functions also given in the proofs section.
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==Derivatives of trigonometric functions and their inverses==
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:<math> \left(\sin(x)\right)' = \cos(x)</math>
:<math> \left(\cos(x)\right)' = -\sin(x)</math>
:<math> \left(\tan(x)\right)' = \left(\frac{\sin(x)}{\cos(x)}\right)' = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)} = \sec^2(x)</math>
:<math> \left(\cot(x)\right)' = \left(\frac{\cos(x)}{\sin(x)}\right)' = \frac{-\sin^2(x) - \cos^2(x)}{\sin^2(x)} = -(1+\cot^2(x)) = -\csc^2(x)</math>
:<math> \left(\sec(x)\right)' = \left(\frac{1}{\cos(x)}\right)' = \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)}.\frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x)</math>
:<math> \left(\csc(x)\right)' = \left(\frac{1}{\sin(x)}\right)' = -\frac{\cos(x)}{\sin^2(x)} = -\frac{1}{\sin(x)}.\frac{\cos(x)}{\sin(x)} = -\csc(x)\cot(x)</math>
:<math> \left(\arcsin(x)\right)' = \frac{1}{\sqrt{1-x^2}}</math>
:<math> \left(\arccos(x)\right)' = \frac{-1}{\sqrt{1-x^2}}</math>
:<math> \left(\arctan(x)\right)' = \frac{1}{x^2+1}</math>


==Proofs of derivatives of trigonometric functions==
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===Limit of <math>\frac{\sin\theta}{\theta}</math> as <math>\theta \to 0</math>===
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[[File:limit_circle_FbN.jpeg|thumb|<center>Circle, centre ''O'', radius ''r''</center>]]


The diagram on the right shows a circle, centre ''O'' and radius ''r''. Let θ be the angle at ''O'' made by the two radii ''OA'' and ''OB''. Since we are considering the limit as θ tends to zero, we may assume that θ is a very small positive number: {{nowrap|1=0 < θ ≪ 1}}.
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Consider the following three regions of the diagram: ''R''<sub>1</sub> is the triangle ''OAB'', ''R''<sub>2</sub> is the [[circular sector]] ''OAB'', and ''R''<sub>3</sub> is the triangle ''OAC''. Clearly:
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:<math>\text{Area}(R_1) < \text{Area}(R_2) < \text{Area}(R_3) \, . </math>
Using basic trigonometric formulae, the area of the triangle ''OAB'' is
:<math> \frac{1}{2} \times ||OA|| \times ||OB|| \times \sin\theta = \frac{1}{2}r^2\sin\theta \, . </math>
The [[Circular_sector#Area|area of the circular sector]] ''OAB'' is <math>\frac{1}{2}r^2\theta</math>, while the area of the triangle ''OAC'' is given by
:<math> \frac{1}{2} \times ||OA|| \times ||AC|| = \frac{1}{2}\times r \times r \tan\theta = \frac{1}{2}r^2\tan\theta \, . </math>
Collecting together these three areas gives:
:<math>\text{Area}(R_1) < \text{Area}(R_2) < \text{Area}(R_3) \iff
\frac{1}{2}r^2\sin\theta < \frac{1}{2}r^2\theta < \frac{1}{2}r^2\tan\theta \, . </math>
Since {{nowrap|1=''r'' > 0}} we can divide through by ½·r<sup>2</sup>; this means that the construction and calculations are all independent of the circle's radius. Moreover, since {{nowrap|1=0 < θ ≪ 1}} it follows that {{nowrap|1=sin(θ) > 0}} and we may divide through by a factor of sin(θ), giving:
:<math>1 < \frac{\theta}{\sin\theta} < \frac{1}{\cos\theta} \implies 1 > \frac{\sin\theta}{\theta} > \cos\theta \, . </math>
In the last step we simply took the reciprocal of each of the three terms. Since all three terms are positive this has the effect of reversing the inequities, e.g. if {{nowrap|1=2 < 3}} then {{nowrap|1=½ > ⅓}}.


[[File:Squeeze FbN.png|thumb|<center>Squeeze: The curves {{nowrap|1=''y'' = 1}} and {{nowrap|1=''y'' = cos(θ)}} shown in red, the curve {{nowrap|1=''y'' = sin(θ)/θ}} shown in blue.</center>]]
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We have seen that if {{nowrap|1=0 < θ ≪ 1}} then sin(θ)/θ is ''always'' less than 1 and, in addition, is ''always'' greater than cos(θ). Notice that as θ gets closer to 0, so cos(θ) gets closer to 1. Informally: as θ gets smaller, sin(θ)/θ is "[[Squeeze theorem|squeezed]]" between 1 and cos(θ), which itself it heading towards 1. It follows that sin(θ)/θ tends to 1 as θ tends to 0 from the positive side.
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For the case where θ is a very small negative number: {{nowrap|1=–1 ≪ θ < 0}}, we use the fact that sine is an [[odd function]]:
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:<math>\lim_{\theta \to 0^-} \frac{\sin\theta}{\theta} = \lim_{\theta\to 0^+}\frac{\sin(-\theta)}{-\theta} = \lim_{\theta \to 0^+}\frac{-\sin\theta}{-\theta} = \lim_{\theta\to 0^+}\frac{\sin\theta}{\theta} = 1 \, . </math>
 
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===Limit of <math>\frac{\cos\theta - 1}{\theta}</math> as <math>\theta \to 0</math>===
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The last section enables us to calculate this new limit relatively easily. This is done by employing a simple trick. In this calculation, the sign of θ is unimportant.
:<math> \lim_{\theta \to 0} \left(\frac{\cos\theta - 1}{\theta}\right) = \lim_{\theta \to 0} \left[ \left( \frac{\cos\theta - 1}{\theta} \right) \left( \frac{\cos\theta + 1}{\cos\theta + 1} \right) \right] = \lim_{\theta \to 0} \left( \frac{\cos^2\theta - 1}{\theta(\cos\theta + 1)} \right) . </math>
The well-known identity {{nowrap|1=sin<sup>2</sup>θ + cos<sup>2</sup>θ = 1}} tells us that {{nowrap|1=cos<sup>2</sup>θ – 1 = –sin<sup>2</sup>θ.}}
Using this, the fact that the limit of a product is the product of the limits, and the result from the last section, we find that:
:<math> \lim_{\theta \to 0} \left(\frac{\cos\theta - 1}{\theta}\right) = \lim_{\theta \to 0} \left( \frac{-\sin^2\theta}{\theta(\cos\theta+1)} \right) = \lim_{\theta \to 0} \left( \frac{-\sin\theta}{\theta}\right) \times \lim_{\theta \to 0} \left( \frac{\sin\theta}{\cos\theta + 1} \right) = (-1) \times \frac{0}{2} = 0 \, . </math>


===Limit of <math>\frac{\tan\theta}{\theta}</math> as <math>\theta \to 0</math>===
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Using the limit for the [[#Limit_of_as|sine]] function, the fact that the tangent function is odd, and the fact that the limit of a product is the product of the limits, we find:
 
:<math>
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\lim_{\theta \to 0^-} \frac{\tan\theta}{\theta}
 
= \lim_{\theta\to 0^+} \frac{\tan\theta}{\theta}
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= \lim_{\theta \to 0} \frac{\tan\theta}{\theta}
 
= \lim_{\theta\to 0} \frac{\sin\theta}{\theta} \times \lim_{\theta\to 0} \frac{1}{\cos\theta}
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= 1 \times 1
 
  = 1 \, . </math>
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===Derivative of the sine function===
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To calculate the derivative of the [[sine function]] sin ''&theta;'', we use [[Derivative#Definition_via_difference_quotients|first principles]]. By definition:
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = \lim_{\delta \to 0} \left( \frac{\sin(\theta + \delta) - \sin \theta}{\delta} \right) . </math>
Using the well-known angle formula {{nowrap|1=sin(α+β) = sin α cos β + sin β cos α}}, we have:
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = \lim_{\delta \to 0} \left( \frac{\sin\theta\cos\delta + \sin\delta\cos\theta-\sin\theta}{\delta} \right) = \lim_{\delta \to 0} \left[ \left(\frac{\sin\delta}{\delta} \cos\theta\right) + \left(\frac{\cos\delta -1}{\delta}\sin\theta\right) \right] . </math>
Using the limits for the [[#Limit_of_as|sine]] and [[#Limit_of_as_2|cosine]] functions:
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = (1\times\cos\theta) + (0\times\sin\theta) = \cos\theta \, . </math>


===Derivative of the cosine function===
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To calculate the derivative of the [[cosine function]] cos ''&theta;'', we use [[Derivative#Definition_via_difference_quotients|first principles]]. By definition:
 
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = \lim_{\delta \to 0} \left( \frac{\cos(\theta+\delta)-\cos\theta}{\delta} \right) . </math>
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Using the well-known angle formula {{nowrap|1=cos(α+β) = cos α cos β – sin α sin β}}, we have:
 
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = \lim_{\delta \to 0} \left( \frac{\cos\theta\cos\delta - \sin\theta\sin\delta-\cos\theta}{\delta} \right) = \lim_{\delta \to 0} \left[ \left(\frac{\cos\delta -1}{\delta}\cos\theta\right) - \left(\frac{\sin\delta}{\delta} \sin\theta\right) \right] . </math>
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Using the limits for the [[#Limit_of_as|sine]] and [[#Limit_of_as_2|cosine]] functions:
 
:<math> \frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = (0 \times \cos\theta) - (1 \times \sin\theta) = -\sin\theta \, . </math>
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===Derivative of the tangent function===
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To calculate the derivative of the [[tangent function]] tan ''&theta;'', we use [[Derivative#Definition_via_difference_quotients|first principles]]. By definition:
:<math>
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta
= \lim_{\delta \to 0} \left( \frac{\tan(\theta+\delta)-\tan\theta}{\delta} \right) .
</math>
Using the well-known angle formula {{nowrap|1=tan(α+β) = (tan α + tan β) / (1 - tan α tan β)}}, we have:
:<math>
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta
  = \lim_{\delta \to 0} \left[ \frac{\frac{\tan\theta + \tan\delta}{1 - \tan\theta\tan\delta} - \tan\theta}{\delta} \right]
= \lim_{\delta \to 0} \left[ \frac{\tan\theta + \tan\delta - \tan\theta + \tan^2\theta\tan\delta}{\delta \left( 1 - \tan\theta\tan\delta \right)} \right] .
</math>
Using the fact that the limit of a product is the product of the limits:
:<math>
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta
= \lim_{\delta \to 0} \frac{\tan\delta}{\delta} \times \lim_{\delta \to 0} \left( \frac{1 + \tan^2\theta}{1 - \tan\theta\tan\delta} \right) .
</math>
Using the limit for the [[#Limit_of_as_3|tangent]] function, and the fact that tan ''&delta;'' tends to 0 as &delta; tends to 0:
:<math>
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta
= 1 \times \frac{1 + \tan^2\theta}{1 - 0} = 1 + \tan^2\theta .
</math>
We see immediately that:
:<math>
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta
= 1 + \frac{\sin^2\theta}{\cos^2\theta}
= \frac{\cos^2\theta + \sin^2\theta}{\cos^2\theta}
= \frac{1}{\cos^2\theta}
= \sec^2\theta \, .
</math>


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The following derivatives are found by setting a [[Variable (mathematics)|variable]] ''y'' equal to the [[inverse trigonometric function]] that we wish to take the derivative of. Using [[implicit differentiation]] and then solving for ''dy''/''dx'', the derivative of the inverse function is found in terms of ''y''. To convert ''dy''/''dx'' back into being in terms of ''x'', we can draw a reference triangle on the unit circle, letting ''θ'' be y. Using the [[Pythagorean theorem]] and the definition of the regular trigonometric functions, we can finally express ''dy''/''dx'' in terms of ''x''.
 
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===Differentiating the inverse sine function===
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:<math>y=\arcsin x\,\!</math>
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:<math>{d \over dx}\tan y={d \over dx}x</math>
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{d \over dx}\tan y
= {d \over dx}\frac{\sin y}{\cos y}
= \frac{{dy \over dx} \cos^2 y + \sin^2 y {dy \over dx}}{\cos^2 y}
= {dy \over dx} \left (1 + \tan^2 y \right)
</math>


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:<math>{d \over dx}x = 1</math>
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Substituting <math>x=\tan y\,\!</math> in from above, we get
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:<math>{dy \over dx}(1+x^2)=1</math>
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:<math>{dy \over dx}=\frac{1}{1+x^2}</math>
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===Differentiating the inverse cotangent function===
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We let
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:<math>y=\arccot x\,\!</math>
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Where
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:<math> 0 < y < \pi</math>
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Then
</ul>
 
:<math>\cot y=x\,\!</math>
 
Using implicit differentiation and solving for dy/dx:
 
:<math>{d \over dx}\cot y={d \over dx}x</math>
 
:<math>{dy \over dx}-\csc^2 y=1</math>
 
Substituting <math>1+\cot^2 y = \csc^2 y\,\!</math> into the above,
 
:<math>{dy \over dx}-(1+\cot^2 y)=1</math>
 
Substituting <math>x=\cot y\,\!</math> in from above, we get
 
:<math>{dy \over dx}-(1+x^2)=1</math>
 
:<math>{dy \over dx}=\frac{-1}{1+x^2}</math>
 
==See also==
*[[Trigonometry]]
*[[Calculus]]
*[[Derivative]]
*[[Table of derivatives]]
 
==References==
{{Reflist}}
 
==Bibliography==
*''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', Edited by Abramowitz and Stegun, National Bureau of Standards, Applied Mathematics Series, 55 (1964)
 
{{DEFAULTSORT:Differentiation Of Trigonometric Functions}}
[[Category:Differential calculus]]

Latest revision as of 07:56, 10 November 2014

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