Trigonometric Equations, Inequalities & Identities (College Board AP® Precalculus): Exam Questions

The function $k$ is given by $k\left(x\right)=\frac{6}{\tan x\left({\csc }^{2}x-1\right)}$.

Rewrite $k\left(x\right)$ as an expression in which $\tan x$ appears exactly once and no other trigonometric functions are involved.

The function $k$ is given by $k\left(x\right)=\left(\frac{1-{\sin }^{2}x}{\sin x}\right)\sec x$.

Rewrite $k\left(x\right)$ as a single term involving $\tan x$.

The function $m$ is given by

$m\left(x\right)={\cos }^{-1}\left(\tan \left(2x\right)\right)$.

Find all values in the domain of $m$ that yield an output value of 0.

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

Solve $h\left(x\right)=\frac{\pi }{4}$ for values of $x$ in the domain of $h$.

Find the exact value of $\sin \left(\frac{7\pi }{12}\right)$.

Find all values of $x$ in the interval $\left[0,2\pi \right]$ for which $\cos x\le -\frac{1}{2}$.

Find all values of $x$ in the interval $\left[0,2\pi \right]$ for which ${\cos }^{2}x-{\sin }^{2}x=1$.

The function $h$ is given by

$h\left(x\right)=\frac{1-{\cos }^{2}x}{\sin x}$

Rewrite $h\left(x\right)$ as an expression in which $\sin x$ appears once and no other trigonometric functions are involved.

The function $k$ is given by $k\left(x\right)=\left(2+2{\tan }^{2}x\right)\cos x$.

Rewrite $k\left(x\right)$ as an expression in which $\sec x$ appears exactly once and no other trigonometric functions are involved.

The function $k$ is given by $k\left(x\right)=\frac{{\csc }^{2}x-1}{\cos x·\csc x}$

Rewrite $k\left(x\right)$ as an expression in which $\cot x$ appears exactly once and no other trigonometric functions are involved. Show the work that leads to your answer.

The function $p$ is given by $p\left(x\right)=2{\sec }^{2}x-4$.

Solve $p\left(x\right)=0$ for values of $x$ in the interval $\left[0,2\pi \right)$. Show the computations that lead to your answer.

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

Solve $h\left(x\right)=\frac{\pi }{2}$ for values of $x$ in the domain of $h$.

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

Solve $h\left(x\right)=\frac{2\pi }{3}$ for values of $x$ in the domain of $h$. Show the computations that lead to your answer.

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

Solve $j\left(x\right)=0$ for values of $x$ in the interval $\left[0,\frac{\pi }{2}\right]$.

The function $m$ is given by $m\left(x\right)=\sin \left(2x\right)+3$. Find all input values in the domain of $m$ that yield an output value of $\frac{7}{2}$.

Find all values of $x$ in the interval $\left[0,2\pi \right]$ for which $\sin \left(2x\right)=\sin \left(x\right)$.