Inverse & Reciprocal Trigonometric Functions (College Board AP® Precalculus): Exam Questions

Find the exact value of each of the following.

(i) ${\sin }^{-1}\left(\frac{1}{2}\right)$

(ii) ${\cos }^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

(iii) $\arctan \left(-\sqrt{3}\right)$

Rewrite the expression $\frac{\sin x}{\csc x}$ as an expression in which $\sin x$ appears exactly once and no other trigonometric functions are involved.

The function $m$ is given by $m\left(x\right)=\sin \left(\arctan x\right)$.

Determine the exact value of $m\left(1\right)$. Show the work that leads to your answer.

The function $f$ is given by $f\left(x\right)=3\arcsin \left(\frac{x}{2}\right)$.

Determine the domain and the range of $f$. Give a mathematical rationale for both based on the domain and range of the parent function $y=\arcsin x$.

The function $g$ is given by $g\left(x\right)=-2\sec x$.

(i) State the range of $g$.

(ii) Determine whether the point $(0,-2)$ is a local minimum or a local maximum of $g$. Give a reason for your answer based on the behavior of the cosine function.

An angle $\theta$ in standard position satisfies $\frac{\pi }{2}<\theta <\pi$, and $\sec \theta =-\frac{5}{4}$.

(i) Find the exact value of $\cos \theta$. Show the work that leads to your answer.

(ii) Find the exact value of $\sin \theta$. Show the work that leads to your answer.

(iii) Find the exact value of $\cot \theta$.

The function $f$ is given by $f\left(x\right)=2\arctan \left(x\right)+1$.

Find the value of $f\left(3\right)$, expressed as a decimal approximation. Show the work that leads to your answer.

Find the limiting value of $f\left(x\right)$ as $x\to \infty$. Give an exact answer. Show the work that leads to your answer.

Rewrite the expression $\frac{\tan x+\cot x}{\sec x\csc x}$ as a single constant.

The function $g$ is given by $g\left(x\right)=-2\sin \left(x\right)$.

State a restricted domain for $g$ on which the inverse function of $g$ can be defined. Give a reason for your answer.

The function $p$ is given by $p\left(x\right)=3\csc \left(2x\right)$.

Find all vertical asymptotes of the graph of $p$ on the interval $(0,\pi )$. Give a reason for your answer.

Rewrite $\sec \left(\arctan x\right)$ as an expression in $x$ that does not involve any trigonometric or inverse trigonometric functions.

Rewrite the expression $\frac{\cot x}{\csc x+1}+\frac{\csc x+1}{\cot x}$ as a single term involving $\sec x$.