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

What is the value of $x$ in the interval $\left[0,\frac{\pi }{2}\right]$ for which $2\sin x=\sqrt{2}$?

Which of the following is equivalent to $\cos \left(\alpha -\beta \right)$?

The function $h$ is given by $h\left(x\right)=5\cos \left(2x\right)$. Which of the following is an equivalent form for $h\left(x\right)$ ?

What are all values of $\theta$, for $0\le \theta <2\pi$, where $2{\cos }^2\theta =\cos \theta$?

The function $f$ is given by $f\left(x\right)={\cos }^2\left(0.5x+0.3\right)$. The function $g$ is given by $g\left(x\right)=f\left(x\right)-0.5$. What are the zeros of $g$ on the interval $0\le x\le 2\pi$ ?

The function $f$ is given by $f\left(x\right)={\sec }^2\left(0.5x+0.4\right)$. The function $g$ is given by $g\left(x\right)=f\left(x\right)-2$. What are the zeros of $g$ on the interval $0\le x\le 2\pi$ ?

Let $f\left(x\right)=2+4\csc x$ and $g\left(x\right)=-6$. In the $xy$-plane, what are the $x$-coordinates of the points of intersection of the graphs of $f$ and $g$ for $0<x<2\pi$?

For $-\pi <\theta <\pi$, which of the following gives all values of $\theta$ for which $2\cos \theta >-1$ and $2\sin \theta >-\sqrt{3}$?

The function $h$ is given by $h\left(x\right)=3\cos \left(2x\right)$. Which of the following is an equivalent form for $h\left(x\right)$?

What are all values of $\theta$, for $0\le \theta <2\pi$, where $2{\sin }^2\theta =\sin \theta$?

An angle $\theta$ is in standard position. If $\sin \theta =-\frac{2}{3}$ and $\pi <\theta <\frac{3\pi }{2}$, what is the exact value of $\sec \theta$ ?

What is the exact value of $\cos \left(\frac{5\pi }{12}\right)$ ?

The expression $\frac{1+\cos \theta }{\sin \theta }+\frac{\sin \theta }{1+\cos \theta }$ is equivalent to which of the following?

What are all values of $x$, for $0\le x<2\pi$, for which $2\sin x\ge 1$ ?

Which of the following is a value of $x$ in the interval $\left[0,\pi \right]$ for which $\sin \left(2x\right)=\frac{1}{2}$?

For $\sin x\ne 0$ and $\cos x\ne 1$, a student attempts to simplify the expression $\frac{\sin x}{1-\cos x}+\frac{1-\cos x}{\sin x}$ using the following four steps:

Step 1. Combine over a common denominator:

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

Step 2. Expand the squared term in the numerator:

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

Step 3. Apply the Pythagorean identity ${\sin }^{2}x=1-{\cos }^{2}x$:

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

Step 4. Factor $1-{\cos }^{2}x=\left(1-\cos x\right)\left(1+\cos x\right)$ and cancel:

$=\frac{2\left(1+\cos x\right)}{\sin x}$

In which step does the first mathematical error occur?

Suppose $\sin A=\frac{3}{5}$ where $\frac{\pi }{2}<A<\pi$, and $\cos B=\frac{5}{13}$ where $\frac{3\pi }{2}<B<2\pi$. What is the value of $\cos \left(A-B\right)$ ?

What are all values of $x$ in the interval $0\le x<2\pi$ for which $2{\sin }^{2}x+\cos x=1$ ?

If $\sin \theta =\frac{3}{5}$ and $\frac{\pi }{2}<\theta <\pi$, what is the value of $\sin \left(2\theta \right)$?

How many values of $x$ in the interval $\left[0,2\pi \right)$ satisfy $2\sin \left(2x-\frac{\pi }{3}\right)=\sqrt{3}$?

How many values of $x$ in the interval $\left[0,2\pi \right)$ satisfy $\sec x+\tan x=1$?