AP Calculus AB and BC glossary

Inverse trigonometric function

Also called: Inverse trig function

An inverse trigonometric function undoes a trig function on a domain restricted to make it one to one. Arcsine returns angles from negative pi over two to pi over two, arccosine returns angles from zero to pi, and arctangent returns angles strictly between negative pi over two and pi over two.

\sin y = x \;\Rightarrow\; \cos y \cdot \frac{dy}{dx} = 1 \;\Rightarrow\; \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^{2}}}

Sine takes the value $\frac{1}{2}$ at infinitely many angles, so it has no inverse until the domain is cut down. The agreed cut keeps $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$ for sine and $0 \le \theta \le \pi$ for cosine, which is why $\arcsin\left(\sin\frac{5\pi}{6}\right)$ is $\frac{\pi}{6}$ and not $\frac{5\pi}{6}$ .

Writing $y = \arcsin x$ as $\sin y = x$ and differentiating implicitly gives $\frac{dy}{dx} = \frac{1}{\cos y}$ , and the Pythagorean identity turns $\cos y$ into $\sqrt{1 - x^{2}}$ . The positive root is correct only because the restricted range keeps $\cos y \ge 0$ , so the domain restriction is what fixes the sign. The algebraic result is why these functions show up as standard antiderivatives, as in $\int \frac{1}{1 + x^{2}}\,dx = \arctan x + C$ .

The mistake

Reading $\sin^{-1}x$ as $\frac{1}{\sin x}$ . The $-1$ marks an inverse function, while the 2 in $\sin^{2}x$ marks a power, so one notation means two different things. The reciprocal of $\sin x$ is $\csc x$ , a completely different function with a completely different derivative.

Appears in: Unit 3: Chain Rule, Implicit, and Inverses, Unit 6: Integration and Accumulation