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Section 2.8 Derivatives of Trigonometric Functions

We are now going to compute the derivatives of the various trigonometric functions, \(\sin x\text{,}\) \(\cos x\) and so on. The computations are more involved than the others that we have done so far and will take several steps. Fortunately, the final answers will be very simple.

Observe that we only need to work out the derivatives of \(\sin x\) and \(\cos x\text{,}\) since the other trigonometric functions are really just quotients of these two functions. Recall:

\begin{align*} \tan x &= \frac{\sin x}{\cos x} & \cot x &= \frac{\cos x}{\sin x} & \csc x &= \frac{1}{\sin x} & \sec x &= \frac{1}{\cos x}. \end{align*}

The first steps towards computing the derivatives of \(\sin x, \cos x\) is to find their derivatives at \(x=0\text{.}\) The derivatives at general points \(x\) will follow quickly from these, using trig identities. It is important to note that we must measure angles in radians  1 In science, radians is the standard unit for measuring angles. While you may be more familiar with degrees, radians should be used in any computation involving calculus. Using degrees will cause errors. Thankfully it is easy to translate between these two measures since \(360^\circ = 2\pi\) radians. See Appendix B.2.1. , rather than degrees, in what follows. Indeed — unless explicitly stated otherwise, any number that is put into a trigonometric function is measured in radians.