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

Which of the following angles has the same terminal ray in standard position as an angle of measure $-\frac{3\pi }{4}$ radians?

The graph of $y=\sin x$ for $0\le x\le 2\pi$ is shown.

Based on the graph, on which of the following intervals is $\sin x$ decreasing?

What is the exact value of $\sin \left(\frac{11\pi }{6}\right)$?

The function $f$ is given by $f\left(x\right)=\tan x$.

For which of the following values of $x$ does the graph of $f$ have a vertical asymptote?

The figure shows a circle centered at the origin with an angle of measure $\theta$ radians in standard position. The terminal ray of the angle intersects the circle at point $P$, and point $Q$ also lies on the circle. The coordinates of $P$ are $\left(x,y\right)$, and $Q$ is a reflection of $P$ in the $y$-axis. Which of the following gives the $x$-coordinate of $Q$?

The figure shows a circle of radius 4 centered at the origin O. The point P lies in Quadrant I on the circle, where the angle from the positive x-axis to the ray OP is $\theta$. The point Q is the reflection of P across the x-axis.

Which of the following gives the y-coordinate of Q?

A circle is centered at the origin with radius $4$. A point $P$ starts at $\left(4,0\right)$ and travels a distance of $6$ units along the circle in the counterclockwise direction. Which of the following gives the coordinates of $P$ after this movement?

The function $g$ is given by $g\left(x\right)=2\tan \left(\frac{x}{3}\right)+1$. What is the period of $g$?

The scatterplot shows values of a periodic function $f$ at integer values of $t$ for $0\le t\le 10$.

Based on the values shown, what is the period of $f$?

The function $g$ is given by $g\left(x\right)=\tan \left(2x-\frac{\pi }{3}\right)$. At which of the following values of $x$ for $0\le x\le \frac{\pi }{2}$ does the graph of $g$ have a vertical asymptote?

The terminal ray of an angle $\theta$ in standard position passes through the point $P$ on the unit circle. If $\sin \theta =-\frac{1}{2}$ and $\theta$ is in the third quadrant, what is the value of $\tan \theta$?

What is the exact value of $\sin \left(\frac{2\pi }{3}\right)-\cos \left(\frac{5\pi }{6}\right)$?

A point $P$ on a circle of radius $6$ centered at the origin begins at $\left(6,0\right)$. $P$ first moves clockwise along the circle through an arc of length $5$ units, then moves counterclockwise along the circle through an arc of length $14$ units. What is the $y$-coordinate of the final position of $P$, expressed as a decimal approximation?

The function $f$ is given by $f\left(x\right)=\cos x$.

On which of the following intervals is $f\left(x\right)$ both negative and increasing?

The function $g$ is periodic with a period of $4$. The table gives values of $g$ for selected values of $x$.

What is the average rate of change of $g$ on the interval $\left[57,67\right]$?

Let $f$ be the function defined by $f\left(x\right)=\tan \left(\frac{\pi }{2}x-\pi \right)$. Let $x_1$ and $x_2$ be the two smallest positive values of $x$ at which $f$ has a vertical asymptote. What is the value of $x_1+x_2$?

The angle $\theta$ is in quadrant III and satisfies $\sin \theta =-\frac{2}{3}$. What is the value of $\tan \theta$?

An angle $\theta$ in standard position has its terminal side passing through the point $\left(-5,12\right)$. What is the value of $\sin \left(\theta +\pi \right)$?