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

The angle $\theta =\frac{2\pi }{3}$ is in standard position.

(i) Find the exact value of $\sin \theta$ by using an appropriate reference angle in the interval $0\le \theta \le \frac{\pi }{2}$. Show the work that leads to your answer.

(ii) Find the exact value of $\cos \theta$ by using an appropriate reference angle in the interval $0\le \theta \le \frac{\pi }{2}$. Show the work that leads to your answer.

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

Consider the tangent function $f$ given by $f\left(\theta \right)=\tan \left(\theta \right)$ on the open interval $\left(0,\frac{\pi }{2}\right)$.

(i) Explain why $\tan \left(\theta \right)>0$ for all $\theta$ in the interval $\left(0,\frac{\pi }{2}\right)$.

(ii) Is $f$ strictly increasing or strictly decreasing on $\left(0,\frac{\pi }{2}\right)$? Give a reason for your answer using the definition of tangent in terms of sine and cosine.

A circle is centered at the origin with radius $4$. The point $P=\left(-2\sqrt{3},2\right)$ lies on the circle at an angle of $\frac{5\pi }{6}$ radians measured counterclockwise from the positive $x$-axis.

(i) Determine the exact value of $\sin \left(\frac{5\pi }{6}\right)$. Show the computations that lead to your answer.

(ii) Determine the exact value of $\cos \left(\frac{5\pi }{6}\right)$. Show the computations that lead to your answer.

The point $P=\left(-3,4\right)$ lies on a circle centered at the origin. The angle $\theta$ in standard position has its terminal ray passing through $P$.

(i) Explain why $\tan \left(\theta \right)$ is equal to the slope of the terminal ray of $\theta t$ for any angle in standard position.

(ii) Use the result from part (i) to find the exact value of $\tan \left(\theta \right)$. Show the computations that lead to your answer.

The function $f$ is periodic with period $7$. Selected values of $f$ are given in the table below.

(i) Find $f\left(23\right)$. Show the computations that lead to your answer.

(ii) Find $f\left(-11\right)$. Show the computations that lead to your answer.

(iii) Find all values of $x$ in the interval $\left[0,21\right]$ for which $f\left(x\right)=6$.

The figure shows a unit circle centered at the origin $O$ in the $xy$-plane. The point $P$ is on the unit circle, and $\theta$ is the angle between the positive $x$-axis and the radius $OP$.

(i) Using the figure, explain what $\sin \left(\theta \right)$ represents geometrically on the unit circle.

(ii) Hence explain why $-1\le \sin \left(\theta \right)\le 1$ for all values of $\theta$.

A point $P$ moves counterclockwise around a circular track of radius $8$ meters centered at the origin. At time $t=0$ seconds, $P$ is at the position $\left(8,0\right)$. $P$ moves along the track at a constant rate of $5$ meters per second.

The angle $\theta$ in standard position has its terminal ray passing through $P$. Find the value of $\theta$, in radians, at time $t=4$ seconds. Show the work that leads to your answer.

Find the coordinates of $P$ at time $t=4$ seconds. Express the coordinates as decimal approximations. Show the work that leads to your answer.

The function $h$ is periodic with a period of $6$. The table gives values for $h\left(x\right)$ at selected values of $x$ within one period.

Find the value of $h\left(50\right)$. Show the work that leads to your answer.

Find the average rate of change of $h$ on the interval $\left[20,23\right]$. Show the work that leads to your answer.

The function $g$ is given by $g\left(x\right)=-3\tan \left(\frac{x}{4}\right)$.

(i) Determine the period of $g$. Show the computations that lead to your answer.

(ii) State whether $g$ is strictly increasing or strictly decreasing on any open interval between consecutive vertical asymptotes. Explain why, using the transformations applied to the parent tangent function $\tan x$.

(iii) Explain why the parent tangent function $\tan x$ has vertical asymptotes, referencing the sine and cosine functions.

The function $g$ is given by $g\left(x\right)=a\tan \left(bx\right)+d$, where $a$, $b$, and $d$ are constants with $a>0$ and $b>0$. The graph of $g$:

• has consecutive vertical asymptotes at $x=-\frac{\pi }{6}$ and $x=\frac{\pi }{6}$

• passes through the points $\left(0,1\right)$ and $\left(\frac{\pi }{12},7\right)$

Determine the value of $b$. Show the work that leads to your answer.

Determine the value of $d$. Show the work that leads to your answer.

Determine the value of $a$. Show the work that leads to your answer.

The graph of a periodic function $f$ is shown above for $0\le x\le 7$. The function $f$ is defined for all real numbers and has period $3$.

Find the value of $f\left(100\right)$. Show the work that leads to your answer.

Find the average rate of change of $f$ on the interval $13\le x\le 15$. Show the work that leads to your answer.

On the interval $\left(43,44\right)$, describe the concavity of the graph of $f$ and determine whether the rate of change of $f$ is increasing or decreasing.