Section Summary

In this section we have added the tangent function to our list of important trigonometric functions — the list now includes sine, cosine, and tangent. While circular motion led us naturally to the sine and cosine, the tangent (ratio of sine to cosine) turned up in the solution of a problem in right-triangle trigonometry. We found that the derivative of the tangent function could be expressed as 1 + the square of the same function. However, that derivative has a simpler formula if we bring in a fourth trigonometric function, the secant (reciprocal of the cosine function):

\(\frac{d}{d \theta}\tan\theta = \sec^2\,\theta\).

(We will take up the derivative of the secant, as well as the derivatives of the two remaining trigonometric functions, in the Exercises.)

Next we considered the inverse tangent (arctangent) and inverse sine (arcsine) functions, again motivated by a problem in right-triangle trigonometry. In general, the inverses of periodic functions are not functions, but there is a natural way to restrict the domains of the sine and tangent so that each output value occurs exactly once, and the inverses of these restricted-domain functions are, in fact, functions. Using our tools for differentiating inverse functions, we found that

\(\frac{d}{dx}\tan^{-1} x=\frac{1}{1+x^2}\)

and

\(\frac{d}{dx}\sin^{-1} x=\frac{1}{\sqrt{1-x^2}}\).

The remarkable thing about these two formulas is that the derivatives of these inverse trigonometric functions are algebraic functions. We saw a similar result with the derivative of the natural logarithm function.

Since we have declared sine, cosine, and tangent to be "important," you may reasonably wonder why we haven't said anything about the inverse cosine. In the Exercises we will explore why this function is less interesting and less useful than the inverse sine.