Monday 9 March 2015


The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin(x), cos(x) and tan(x). For example, the derivative of f(x) = sin(x) is represented as f ′(a) = cos(a). f ′(a) is the rate of change of sin(x) at a particular point a.

All derivatives of circular trigonometric functions can be found using those of sin(x) and cos(x) since they can all be expressed in terms of sine or cosine. The quotient rule is then implemented to differentiate the resulting expression. Finding the derivatives of the inverse trigonometric functions involves using implicit differentiation and the derivatives of regular trigonometric functions.

Derivatives of trigonometric functions

Derivative of the sine function

To calculate the derivative of the sine function sin θ, we use first principles. By definition:
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = \lim_{\delta \to 0} \left( \frac{\sin(\theta + \delta) - \sin \theta}{\delta} \right) .
Using the well-known angle formula sin(α+β) = sin α cos β + sin β cos α, we have:
\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] .
Using the limits for the sine and cosine functions:
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\sin\theta = (1\times\cos\theta) + (0\times\sin\theta) = \cos\theta \, .

Derivative of the cosine function

To calculate the derivative of the cosine function cos θ, we use first principles. By definition:
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = \lim_{\delta \to 0} \left( \frac{\cos(\theta+\delta)-\cos\theta}{\delta} \right) .
Using the well-known angle formula cos(α+β) = cos α cos β – sin α sin β, we have:
\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] .
Using the limits for the sine and cosine functions:
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\cos\theta = (0 \times \cos\theta) - (1 \times \sin\theta) = -\sin\theta \, .

From the chain rule

To compute the derivative of the cosine function from the chain rule, first observe the following two facts:
\cos\theta = \sin\left(\theta + \frac{\pi}{2}\right)
\frac{\operatorname{d}}{\operatorname{d}\!\theta} \sin\theta = \cos\theta
The first is a known trigonometric identity, and the second is proven above. Using these two facts, we can write the following,
\frac{\operatorname{d}}{\operatorname{d}\!\theta} \cos\theta = \frac{\operatorname{d}}{\operatorname{d}\!\theta} \sin\left(\theta + \frac{\pi}{2}\right)
We can differentiate this using the chain rule:
\mbox{Let}\ f\!\left(x\right) = \sin x,\ g\!\left(x\right) = \theta + \frac{\pi}{2}
\frac{\operatorname{d}}{\operatorname{d}\!\theta} f\!\left(g\!\left(x\right)\right) = f^\prime\!\left(g\!\left(x\right)\right) \cdot g^\prime\!\left(x\right) = \cos\left(\theta + \frac{\pi}{2}\right) \cdot (1 + 0) = \cos\left(\theta + \frac{\pi}{2}\right)
But, from above, we can rewrite this as
\cos\left(\theta + \frac{\pi}{2}\right) = \sin\left(\left(\theta + \frac{\pi}{2}\right) + \frac{\pi}{2}\right) = \sin\left(\theta + \pi\right)
But, this is an identity relating to horizontally translating the sine function, and therefore,
\sin\left(\theta + \pi\right) = -\sin\theta
Therefore, we have proven that
\frac{\operatorname{d}}{\operatorname{d}\!\theta} \cos\theta = -\sin\theta.

Derivative of the tangent function

To calculate the derivative of the tangent function tan θ, we use first principles. By definition:
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta = \lim_{\delta \to 0} \left( \frac{\tan(\theta+\delta)-\tan\theta}{\delta} \right) .
Using the well-known angle formula tan(α+β) = (tan α + tan β) / (1 - tan α tan β), we have:
\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] .
Using the fact that the limit of a product is the product of the limits:
\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) .
Using the limit for the tangent function, and the fact that tan δ tends to 0 as δ tends to 0:
\frac{\operatorname{d}}{\operatorname{d}\!\theta}\,\tan\theta = 1 \times \frac{1 + \tan^2\theta}{1 - 0} = 1 + \tan^2\theta .
We see immediately that:
\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 \, .

From the quotient rule[edit]

One can also compute the derivative of the tanget function using the quotient rule.
\frac{\operatorname{d}}{\operatorname{d}\!\theta} \tan\theta = \frac{\operatorname{d}}{\operatorname{d}\!\theta} \frac{\sin\theta}{\cos\theta} = \frac{\left(\sin\theta\right)^\prime \cdot \cos\theta - \sin\theta \cdot \left(\cos\theta\right)^\prime}{ \cos^2 \theta } = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}
The numerator can be simplified to 1 by the Pythagorean identity, giving us,
\frac{1}{\cos^2 \theta} = \sec^2 \theta
Therefore,
\frac{\operatorname{d}}{\operatorname{d}\!\theta} \tan\theta = \sec^2 \theta

I also include here a simple video presentation link from Youtube in differentiating trigonometric function using chain rule:




References:
http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions
https://www.youtube.com/watch?v=nZMV2U6BWNE
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