AP Precalculus Unit 3: Trigonometric and Polar Functions

Periodic behavior; sine, cosine and tangent from the unit circle and right triangles; exact values; the graphs of sine and cosine and the sinusoidal form $a\sin(b(x+c))+d$; building sinusoidal models; the tangent function; inverse trigonometric functions; solving trigonometric equations and inequalities; secant, cosecant and cotangent; identities (Pythagorean, sum and difference, double angle); polar coordinates, polar graphs, and how the distance from the pole changes.

CED topics in this unit: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15.

Key formulas and facts

• A function is periodic with period $p$ if $f(x+p)=f(x)$ for all $x$. Radian measure: arc length $s=r\theta$; $\pi$ radians $=180^\circ$.
• On the unit circle the point at angle $\theta$ is $(\cos\theta,\sin\theta)$, and $\tan\theta=\frac{\sin\theta}{\cos\theta}$ is the slope of the terminal ray.
• Exact values at $\frac\pi6,\frac\pi4,\frac\pi3$: $\sin=\frac12,\frac{\sqrt2}2,\frac{\sqrt3}2$ and $\cos=\frac{\sqrt3}2,\frac{\sqrt2}2,\frac12$; use the reference angle and the quadrant's signs for other angles.
• For $f(x)=a\sin(b(x+c))+d$: amplitude $|a|$, period $\frac{2\pi}{|b|}$, phase shift $-c$, midline $y=d$, range $[d-|a|,d+|a|]$.
• Sinusoidal model from data: midline $=\frac{\max+\min}{2}$, amplitude $=\frac{\max-\min}{2}$, and maximum to next minimum is half a period.
• $\tan x$ has period $\pi$ and vertical asymptotes at $x=\frac\pi2+k\pi$; $\tan(bx)$ has period $\frac{\pi}{|b|}$.
• $\arcsin$: domain $[-1,1]$, range $\left[-\frac\pi2,\frac\pi2\right]$. $\arccos$: domain $[-1,1]$, range $[0,\pi]$. $\arctan$: domain all reals, range $\left(-\frac\pi2,\frac\pi2\right)$.
• $\sec x=\frac1{\cos x}$, $\csc x=\frac1{\sin x}$, $\cot x=\frac{\cos x}{\sin x}$.
• Identities: $\sin^2x+\cos^2x=1$, $1+\tan^2x=\sec^2x$, $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$, $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$, $\sin2x=2\sin x\cos x$, $\cos2x=1-2\sin^2x$.
• Polar and rectangular: $x=r\cos\theta$, $y=r\sin\theta$, $r^2=x^2+y^2$, $\tan\theta=\frac yx$ (choose $\theta$ by quadrant).
• Distance from the pole on $r=f(\theta)$ increases where $r$ and its rate of change have the same sign, and decreases where they have opposite signs.

Worked examples

• Exact value: $\cos\frac{7\pi}{4}=\frac{\sqrt2}{2}$ (Quadrant IV, reference angle $\frac\pi4$).
• Sinusoidal features: $y=-4\cos\left(\frac\pi2x\right)+1$ has amplitude $4$, period $4$, midline $y=1$ and range $[-3,5]$.
• Model: a quantity peaks at $16$ when $t=1$ and bottoms out at $4$ when $t=5$: $y=6\cos\left(\frac\pi4(t-1)\right)+10$.
• Equation: $2\sin x=\sqrt3$ on $[0,2\pi)$ gives $x=\frac\pi3$ and $x=\frac{2\pi}{3}$.
• Inequality: $\tan x>1$ on $\left(-\frac\pi2,\frac\pi2\right)$ means $\frac\pi4<x<\frac\pi2$.
• Identity in an equation: $\sin2x=\sin x$ on $[0,2\pi)$ gives $\sin x(2\cos x-1)=0$, so $x=0,\frac\pi3,\pi,\frac{5\pi}3$.
• Inverse trig: $\sin\left(\arccos\frac{5}{13}\right)=\frac{12}{13}$.
• Polar to rectangular: $\left(-4,\frac\pi6\right)$ is the point $\left(-2\sqrt3,-2\right)$.
• Polar rate: for $r=2-2\sin\theta$ on $\left(0,\frac\pi2\right)$, $r>0$ and decreasing, so the curve moves toward the pole; the average rate of change of $r$ there is $\frac{0-2}{\pi/2}=-\frac4\pi$.

Common traps

• Using degrees in a formula that expects radians (or the calculator in the wrong mode).
• Giving $\arcsin\left(\sin\frac{5\pi}6\right)=\frac{5\pi}6$: the answer must lie in $\left[-\frac\pi2,\frac\pi2\right]$.
• Reading the period of $\sin(bx)$ as $b$ instead of $\frac{2\pi}{|b|}$; for tangent it is $\frac\pi{|b|}$.
• Writing the phase shift of $\cos(2x+\pi)$ as $\pi$: factor first, $\cos\left(2\left(x+\frac\pi2\right)\right)$, a shift of $\frac\pi2$ left.
• Dividing both sides of a trig equation by $\sin x$ and losing the solutions where $\sin x=0$.
• Taking $\theta=\arctan\frac yx$ for a point in Quadrant II or III without adding $\pi$.
• Saying the distance from the pole increases because $r$ increases, when $r$ is negative (then $|r|$ decreases).

Practice questions

134 multiple-choice and 4 free-response questions, each tagged with its CED topics and with "No calculator" or "Graphing calculator". Try each one before opening the answer. Each line of a free-response scoring guide is one point, and every free-response question here is out of 6. 35 of the 134 multiple-choice questions here need a graphing calculator; on the May 2027 exam, Part A of the multiple choice is 29 questions with no calculator and Part B is 13 questions with a graphing calculator. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

The table gives values of a periodic function $f$. Based on the table, what is the period of $f$?

1. $2$
2. $8$
3. $3$
4. $4$

Question 2 · Multiple choice

The graph of a periodic function $g$ is shown. What is the period of $g$?

1. $6$
2. $3$
3. $12$
4. $2$

Question 3 · Multiple choice

A function $f$ is periodic with period $5$, and $f(2)=7$. What is $f(17)$?

1. $35$
2. $7$
3. $12$
4. $-7$

Question 4 · Multiple choice

A city's average daily high temperature is modeled as periodic in the day number $t$, with period $365$ days. The model's value on day $400$ equals its value on which day of the first year?

1. Day $400$
2. Day $765$
3. Day $35$
4. Day $36$

Question 5 · Multiple choice

The graph of $f(x)=-2\cos\left(\frac{\pi x}{2}\right)$ is shown for $0\le x\le4$, one full period. On which interval is $f$ increasing and its graph concave down?

1. $(0,1)$
2. $(2,3)$
3. $(0,2)$
4. $(1,2)$

Question 6 · Multiple choice

A periodic function has period $6$ and a maximum at $x=1$. At which of these $x$-values must it also have a maximum?

1. $x=6$
2. $x=4$
3. $x=13$
4. $x=9$

Question 7 · Multiple choice

A function $f$ has period $8$, and $f(x)=|x-4|$ for $0\le x<8$. What is $f(21)$?

1. $17$
2. $1$
3. $5$
4. $3$

Question 8 · Multiple choice

An angle $\theta$ in standard position has its terminal ray through the point $(-3,4)$. What is $\sin\theta$?

1. $\frac{4}{5}$
2. $-\frac{3}{5}$
3. $-\frac{4}{3}$
4. $\frac{3}{4}$

Question 9 · Multiple choice

An angle $\theta$ in standard position has its terminal ray through $(-3,4)$. What is $\tan\theta$?

1. $-\frac{3}{4}$
2. $\frac{4}{3}$
3. $\frac{4}{5}$
4. $-\frac{4}{3}$

Question 10 · Multiple choice

In a right triangle, the side opposite angle $A$ has length $5$ and the hypotenuse has length $13$. What is $\cos A$?

1. $\frac{5}{13}$
2. $\frac{12}{13}$
3. $\frac{5}{12}$
4. $\frac{13}{12}$

Question 11 · Multiple choice

What is $150^\circ$ in radians?

1. $\frac{5\pi}{6}$
2. $\frac{5\pi}{3}$
3. $\frac{3\pi}{4}$
4. $\frac{2\pi}{3}$

Question 12 · Multiple choice

A circle has radius $6$. What is the length of the arc cut off by a central angle of $\frac{2\pi}{3}$ radians?

1. $12\pi$
2. $4$
3. $4\pi$
4. $2\pi$

Question 13 · Multiple choice

A point $P$ is on a circle of radius $10$ centered at the origin, on the terminal ray of the angle $\frac{4\pi}{3}$. What is the $y$-coordinate of $P$?

1. $-5\sqrt{3}$
2. $5\sqrt{3}$
3. $-5$
4. $5$

Question 14 · Multiple choice

An angle $\theta$ is in Quadrant III, and $\cos\theta=-0.6$. What is $\sin\theta$?

1. $0.8$
2. $-0.4$
3. $0.6$
4. $-0.8$

Question 15 · Multiple choice

To three decimal places, how many degrees are in an angle of $1$ radian?

1. $3.142$
2. $57.296$
3. $60.000$
4. $0.017$

Question 16 · Multiple choice

To three decimal places, what is $\sin(2.3)$, where $2.3$ is in radians?

1. $0.040$
2. $-0.666$
3. $-0.746$
4. $0.746$

Question 17 · Multiple choice

An angle of $4$ radians in standard position meets the circle $x^2+y^2=25$ at point $P$. To three decimal places, what is the $x$-coordinate of $P$?

1. $-3.268$
2. $-3.784$
3. $4.988$
4. $3.268$

Question 18 · Multiple choice

Give the exact value of $\sin\left(\frac{11\pi}{6}\right)$.

1. $\frac{1}{2}$
2. $-\frac{\sqrt{3}}{2}$
3. $-\frac{1}{2}$
4. $\frac{\sqrt{3}}{2}$

Question 19 · Multiple choice

What is $\cos\left(\frac{3\pi}{4}\right)$?

1. $\frac{\sqrt{2}}{2}$
2. $-\frac{1}{2}$
3. $-\frac{\sqrt{2}}{2}$
4. $-\frac{\sqrt{3}}{2}$

Question 20 · Multiple choice

What is $\sin\left(\frac{5\pi}{3}\right)+\cos\left(\frac{5\pi}{3}\right)$?

1. $\frac{1+\sqrt{3}}{2}$
2. $\frac{-1-\sqrt{3}}{2}$
3. $\frac{\sqrt{3}-1}{2}$
4. $\frac{1-\sqrt{3}}{2}$

Question 21 · Multiple choice

What is $\tan\left(\frac{5\pi}{4}\right)$?

1. $1$
2. $-1$
3. $\sqrt{3}$
4. $\frac{\sqrt{2}}{2}$

Question 22 · Multiple choice

Which angle has the same sine as $\frac{\pi}{5}$?

1. $\frac{6\pi}{5}$
2. $\frac{4\pi}{5}$
3. $\frac{9\pi}{5}$
4. $-\frac{\pi}{5}$

Question 23 · Multiple choice

Cosine is an even function. Use this to find the exact value of $\cos\left(-\frac{5\pi}{6}\right)$.

1. $-\frac{\sqrt{3}}{2}$
2. $\frac{\sqrt{3}}{2}$
3. $-\frac{1}{2}$
4. $\frac{1}{2}$

Question 24 · Multiple choice

What is $\sin\left(\frac{11\pi}{4}\right)$?

1. $-\frac{\sqrt{2}}{2}$
2. $1$
3. $\frac{\sqrt{2}}{2}$
4. $\frac{1}{2}$

Question 25 · Multiple choice

On $[0,2\pi]$, on which interval is $y=\sin x$ decreasing and concave down?

1. $\left(0,\frac{\pi}{2}\right)$
2. $\left(\frac{\pi}{2},\pi\right)$
3. $\left(\pi,\frac{3\pi}{2}\right)$
4. $\left(\frac{3\pi}{2},2\pi\right)$

Question 26 · Multiple choice

What are the zeros of $y=\cos x$ on $[0,2\pi]$?

1. $\left\{0,\pi,2\pi\right\}$
2. $\left\{\frac{\pi}{2}\right\}$
3. $\left\{\pi\right\}$
4. $\left\{\frac{\pi}{2},\frac{3\pi}{2}\right\}$

Question 27 · Multiple choice

Which function is graphed?

1. $-\sin x$
2. $\sin x$
3. $\cos x$
4. $-\cos x$

Question 28 · Multiple choice

What is the range of $y=\cos x$ on the interval $\left[\frac{\pi}{3},\pi\right]$?

1. $\left[-1,1\right]$
2. $\left[\frac{1}{2},1\right]$
3. $\left[0,\frac{1}{2}\right]$
4. $\left[-1,\frac{1}{2}\right]$

Question 29 · Multiple choice

Which expression equals $\sin(x+\pi)$ for all $x$?

1. $\sin x$
2. $-\sin x$
3. $\cos x$
4. $-\cos x$

Question 30 · Multiple choice

On which interval is $y=\cos x$ increasing?

1. $\left(0,\pi\right)$
2. $\left(\frac{\pi}{2},\frac{3\pi}{2}\right)$
3. $\left(\pi,2\pi\right)$
4. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

Question 31 · Multiple choice

Let $f(x)=\sin x+0.3x$. To three decimal places, at what $x$ in $(0,\pi)$ does $f$ have a relative maximum?

1. $1.571$
2. $1.875$
3. $1.266$
4. $1.517$

Question 32 · Multiple choice

The graphs of $y=\cos x$ and $y=x$ intersect once. To three decimal places, what is the $x$-coordinate of the intersection?

1. $0.785$
2. $1.000$
3. $0.739$
4. $0.540$

Question 33 · Multiple choice

What is the period of $f(x)=3\sin(4x)-1$?

1. $4\pi$
2. $8\pi$
3. $\frac{\pi}{4}$
4. $\frac{\pi}{2}$

Question 34 · Multiple choice

What is the amplitude of $y=-5\cos\left(\frac{x}{2}\right)+2$?

1. $-5$
2. $2$
3. $5$
4. $\frac{5}{2}$

Question 35 · Multiple choice

What is the midline of the graph of $y=4\sin\left(3(x-1)\right)-7$?

1. $y=4$
2. $y=-7$
3. $y=-3$
4. $y=1$

Question 36 · Multiple choice

What is the range of $f(x)=2\cos(\pi x)+3$?

1. $[1,5]$
2. $[-1,1]$
3. $[2,3]$
4. $[-2,5]$

Question 37 · Multiple choice

Let $f(x)=-3\sin(2x)+1$. What is the least positive $x$ at which $f$ has its maximum value?

1. $\frac{\pi}{4}$
2. $\frac{3\pi}{4}$
3. $\frac{3\pi}{2}$
4. $\frac{\pi}{2}$

Question 38 · Multiple choice

Which function has period $6$ and amplitude $2$?

1. $2\sin\left(\frac{\pi x}{3}\right)$
2. $2\sin(6x)$
3. $6\sin(2x)$
4. $2\sin(3\pi x)$

Question 39 · Multiple choice

How many complete cycles does $f(t)=\sin(10\pi t)$ make as $t$ goes from $0$ to $1$?

1. $10$
2. $\frac{1}{5}$
3. $10\pi$
4. $5$

Question 40 · Multiple choice

The function $f(x)=3\sin(2x)+2\cos(2x)$ is sinusoidal. To three decimal places, what is its amplitude?

1. $5.000$
2. $3.000$
3. $2.236$
4. $3.606$

Question 41 · Multiple choice

Let $f(x)=4\cos\big(1.5(x-0.7)\big)+1$. To three decimal places, what is $f(3.1)$?

1. $-2.587$
2. $-1.763$
3. $-3.587$
4. $4.992$

Question 42 · Multiple choice

Let $f(x)=2.5\sin(0.8x)+1.2$. To three decimal places, what is the least positive $x$ for which $f(x)=3$?

1. $2.922$
2. $1.005$
3. $0.804$
4. $0.643$

Question 43 · Multiple choice

The graph of $y=\sin x$ is translated $\frac{\pi}{4}$ units right and then dilated vertically by a factor of $3$. Which is the new function?

1. $3\sin\left(x+\frac{\pi}{4}\right)$
2. $\sin\left(3x-\frac{\pi}{4}\right)$
3. $3\sin\left(x-\frac{\pi}{4}\right)$
4. $3\sin x-\frac{\pi}{4}$

Question 44 · Multiple choice

The function $y=\cos\left(2x+\frac{\pi}{3}\right)$ can be written $y=\cos\left(2(x-h)\right)$. What is $h$?

1. $-\frac{\pi}{6}$
2. $\frac{\pi}{6}$
3. $-\frac{\pi}{3}$
4. $\frac{\pi}{3}$

Question 45 · Multiple choice

The graph of $f(x)=\sin x$ is dilated horizontally by a factor of $3$ and then translated up $2$ units. Which is the new function?

1. $\sin(3x)+2$
2. $\sin\left(\frac{x}{3}\right)+2$
3. $3\sin x+2$
4. $\sin\left(\frac{x}{3}+2\right)$

Question 46 · Multiple choice

For which value of $c$ is $\cos x=\sin(x+c)$ for all $x$?

1. $-\frac{\pi}{2}$
2. $\pi$
3. $\frac{\pi}{2}$
4. $0$

Question 47 · Multiple choice

On $[0,2\pi]$, at what value of $x$ does $g(x)=-2\cos x+1$ reach its maximum?

1. $x=0$
2. $x=\frac{\pi}{2}$
3. $x=2\pi$
4. $x=\pi$

Question 48 · Multiple choice

The graph of $y=\sin x$ is reflected over the $x$-axis and then translated $\frac{\pi}{2}$ units left. Which expression equals the new function?

1. $\cos x$
2. $\sin x$
3. $-\cos x$
4. $-\sin x$

Question 49 · Multiple choice

The graph of a sinusoidal function is shown. Which could be its equation?

1. $1-2\cos(2x)$
2. $1+2\cos(2x)$
3. $2-\cos(2x)$
4. $1-2\cos x$

Question 50 · Multiple choice

Let $g(x)=2\sin\big(3(x+0.4)\big)-1$. To three decimal places, what is the least positive zero of $g$?

1. $0.175$
2. $0.575$
3. $0.473$
4. $0.873$

Question 51 · Multiple choice

Let $g(x)=2\sin\big(3(x+0.4)\big)-1$. To three decimal places, what is $g(1)$?

1. $-1.511$
2. $-0.743$
3. $-0.854$
4. $-2.743$

Question 52 · Multiple choice

The temperature in a greenhouse varies sinusoidally between $50^\circ$F and $80^\circ$F over a $24$-hour period, with the maximum at $t=15$ hours after midnight. Which function models the temperature?

1. $15\cos\left(\frac{\pi}{12}(t+15)\right)+65$
2. $30\cos\left(\frac{\pi}{24}(t-15)\right)+65$
3. $15\sin\left(\frac{\pi}{12}(t-15)\right)+65$
4. $15\cos\left(\frac{\pi}{12}(t-15)\right)+65$

Question 53 · Multiple choice

A weight on a spring is $4$ cm above the floor at its lowest, at $t=0$ seconds, and next reaches its highest point, $10$ cm, at $t=0.5$. Which function models its height?

1. $3\cos(2\pi t)+7$
2. $-3\cos(2\pi t)+7$
3. $-3\cos(\pi t)+7$
4. $-6\cos(2\pi t)+4$

Question 54 · Multiple choice

A sinusoidal model has a maximum of $20$ at $t=3$, and its next minimum is $8$ at $t=7$. What is the period of the model?

1. $4$
2. $12$
3. $8$
4. $14$

Question 55 · Multiple choice

The water depth at a dock, in feet, is modeled by $d(t)=2.5\sin\left(\frac{\pi t}{6}\right)+7$, where $t$ is hours after midnight. What is the depth at $t=4.5$?

1. $8.768$
2. $9.500$
3. $7.205$
4. $5.232$

Question 56 · Multiple choice

The hours of daylight on day $t$ of the year are modeled by $D(t)=12+3\sin\left(\frac{2\pi}{365}(t-80)\right)$. On which day is the modeled daylight longest?

1. $t=80$
2. $t=171.25$
3. $t=262.5$
4. $t=353.5$

Question 57 · Multiple choice

A rider's height on a wheel, in meters, is $h(t)=6\sin\left(\frac{\pi t}{5}\right)+10$ at $t$ seconds. What is the first time $t>0$ at which the height is $13$ meters?

1. $\frac{5}{3}$
2. $\frac{25}{6}$
3. $\frac{1}{2}$
4. $\frac{5}{6}$

Question 58 · Multiple choice

The depth of water at a pier, in meters, is modeled by $d(t)=1.8\cos\left(\dfrac{\pi(t-4)}{6.2}\right)+5$, where $t$ is hours after midnight. To three decimal places, what is the depth at $t=9$?

1. $3.523$
2. $6.525$
3. $6.798$
4. $6.246$

Question 59 · Multiple choice

Using the pier model $d(t)=1.8\cos\left(\dfrac{\pi(t-4)}{6.2}\right)+5$, what is the first time after $t=0$, to three decimal places, at which the depth reaches $6$ meters?

1. $5.938$
2. $2.062$
3. $4.000$
4. $1.562$

Question 60 · Multiple choice

A rider's height above the ground on a Ferris wheel, in meters, is $h(t)=25-22\cos\left(\dfrac{2\pi t}{7.5}\right)$, where $t$ is minutes after boarding. During the first full rotation, for how many minutes is the rider higher than $35$ meters? Give your answer to three decimal places.

1. $1.312$
2. $4.876$
3. $2.624$
4. $2.043$

Question 61 · Multiple choice

How long is one period of the graph of $y=\tan(3x)$?

1. $\frac{2\pi}{3}$
2. $\frac{\pi}{3}$
3. $3\pi$
4. $\pi$

Question 62 · Multiple choice

The graph of $y=\tan\left(\frac{x}{2}\right)$ has infinitely many vertical asymptotes. Which of these lines is one of them?

1. $x=\pi$
2. $x=\frac{\pi}{2}$
3. $x=\frac{\pi}{4}$
4. $x=2\pi$

Question 63 · Multiple choice

What is the slope of the terminal ray of the angle $\frac{2\pi}{3}$ in standard position?

1. $\sqrt{3}$
2. $-\frac{\sqrt{3}}{3}$
3. $-\sqrt{3}$
4. $-\frac{1}{2}$

Question 64 · Multiple choice

If $\tan\theta=2$ and $\theta$ is in Quadrant III, what is $\sin\theta$?

1. $\frac{2\sqrt{5}}{5}$
2. $-\frac{\sqrt{5}}{5}$
3. $-2$
4. $-\frac{2\sqrt{5}}{5}$

Question 65 · Multiple choice

Which statement about $y=\tan x$ on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ is true?

1. It is increasing; concave up on $\left(-\frac{\pi}{2},0\right)$ and concave down on $\left(0,\frac{\pi}{2}\right)$
2. It is decreasing; concave down on $\left(-\frac{\pi}{2},0\right)$ and concave up on $\left(0,\frac{\pi}{2}\right)$
3. It is increasing; concave down on $\left(-\frac{\pi}{2},0\right)$ and concave up on $\left(0,\frac{\pi}{2}\right)$
4. It increases on $\left(0,\frac{\pi}{2}\right)$ only

Question 66 · Multiple choice

Which expression equals $\tan(x+\pi)$ wherever both are defined?

1. $\tan x$
2. $-\tan x$
3. $\cot x$
4. $-\cot x$

Question 67 · Multiple choice

What is the zero of $y=2\tan\left(x-\frac{\pi}{4}\right)$ on the interval $\left(-\frac{\pi}{4},\frac{3\pi}{4}\right)$?

1. $-\frac{\pi}{4}$
2. $0$
3. $\frac{\pi}{2}$
4. $\frac{\pi}{4}$

Question 68 · Multiple choice

To three decimal places, what is the solution of $\tan x=4$ in the interval $\left(\pi,\dfrac{3\pi}{2}\right)$?

1. $1.326$
2. $4.957$
3. $3.387$
4. $4.467$

Question 69 · Multiple choice

Let $f(x)=2\tan(0.5x)$. To three decimal places, what is $f(2.6)$?

1. $7.204$
2. $3.602$
3. $-1.203$
4. $0.045$

Question 70 · Multiple choice

What is $\arcsin\left(-\frac{1}{2}\right)$?

1. $\frac{7\pi}{6}$
2. $-\frac{\pi}{6}$
3. $\frac{11\pi}{6}$
4. $-\frac{\pi}{3}$

Question 71 · Multiple choice

What is $\arccos\left(-\frac{\sqrt{2}}{2}\right)$?

1. $-\frac{\pi}{4}$
2. $\frac{5\pi}{4}$
3. $\frac{\pi}{4}$
4. $\frac{3\pi}{4}$

Question 72 · Multiple choice

What is $\arctan\sqrt{3}$?

1. $\frac{\pi}{6}$
2. $\frac{\pi}{3}$
3. $\frac{4\pi}{3}$
4. $\frac{2\pi}{3}$

Question 73 · Multiple choice

What is $\arccos\left(\cos\frac{5\pi}{3}\right)$?

1. $\frac{5\pi}{3}$
2. $-\frac{\pi}{3}$
3. $\frac{\pi}{3}$
4. $\frac{2\pi}{3}$

Question 74 · Multiple choice

What is the domain of $y=\arcsin x$?

1. $[-1,1]$
2. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$
3. $[0,\pi]$
4. $(-\infty,\infty)$

Question 75 · Multiple choice

What is the range of $y=\arctan x$?

1. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$
2. $(0,\pi)$
3. $(-\infty,\infty)$
4. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

Question 76 · Multiple choice

What is $\cos\left(\arctan\frac{3}{4}\right)$?

1. $\frac{4}{5}$
2. $\frac{3}{5}$
3. $\frac{3}{4}$
4. $\frac{5}{4}$

Question 77 · Multiple choice

To three decimal places, what is $\sin^{-1}(0.62)$, in radians?

1. $0.902$
2. $0.669$
3. $38.316$
4. $0.581$

Question 78 · Multiple choice

To three decimal places, what is $\cos^{-1}(-0.35)$, in radians?

1. $1.213$
2. $-0.358$
3. $1.928$
4. $-1.213$

Question 79 · Multiple choice

What are the solutions of $2\cos x+1=0$ on $[0,2\pi)$?

1. $\left\{\frac{\pi}{3},\frac{5\pi}{3}\right\}$
2. $\left\{\frac{2\pi}{3}\right\}$
3. $\left\{\frac{2\pi}{3},\frac{4\pi}{3}\right\}$
4. $\left\{\frac{5\pi}{6},\frac{7\pi}{6}\right\}$

Question 80 · Multiple choice

What are the solutions of $\sin(2x)=0$ on $[0,\pi]$?

1. $\left\{0,\pi\right\}$
2. $\left\{0,\frac{\pi}{2},\pi\right\}$
3. $\left\{\frac{\pi}{2}\right\}$
4. $\left\{0,\frac{\pi}{4},\frac{\pi}{2}\right\}$

Question 81 · Multiple choice

How many solutions does $\tan x=1$ have on $[0,4\pi)$?

1. $2$
2. $1$
3. $8$
4. $4$

Question 82 · Multiple choice

For which $x$ in $[0,2\pi)$ is $\sin x>\frac{1}{2}$?

1. $\left(\frac{\pi}{3},\frac{2\pi}{3}\right)$
2. $\left(\frac{\pi}{6},\frac{5\pi}{6}\right)$
3. $\left(0,\frac{\pi}{6}\right)$
4. $\left(\frac{7\pi}{6},\frac{11\pi}{6}\right)$

Question 83 · Multiple choice

Solve $3\sin x=1$ on $[0,\pi]$. What is the larger solution, to three decimal places?

1. $2.802$
2. $0.340$
3. $2.731$
4. $1.231$

Question 84 · Multiple choice

What are the solutions of $\cos^2x=\frac{1}{4}$ on $[0,2\pi)$?

1. $\left\{\frac{\pi}{3},\frac{5\pi}{3}\right\}$
2. $\left\{\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}\right\}$
3. $\left\{\frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3}\right\}$
4. $\left\{\frac{2\pi}{3},\frac{4\pi}{3}\right\}$

Question 85 · Multiple choice

The equation $2\cos x=0.7$ has two solutions in $[0,2\pi)$. To three decimal places, what is the larger one?

1. $1.213$
2. $4.355$
3. $5.488$
4. $5.070$

Question 86 · Multiple choice

The equation $\sin x=0.2x$ has exactly one positive solution. To three decimal places, what is it?

1. $2.596$
2. $3.142$
3. $1.571$
4. $0.201$

Question 87 · Multiple choice

To three decimal places, what is the least positive solution of $3\cos(2x)=1$?

1. $1.231$
2. $0.615$
3. $2.526$
4. $0.308$

Question 88 · Multiple choice

What is $\sec\left(\frac{\pi}{3}\right)$?

1. $\frac{1}{2}$
2. $\frac{2\sqrt{3}}{3}$
3. $\sqrt{3}$
4. $2$

Question 89 · Multiple choice

What is $\csc\left(\frac{5\pi}{4}\right)$?

1. $-\sqrt{2}$
2. $\sqrt{2}$
3. $-\frac{\sqrt{2}}{2}$
4. $1$

Question 90 · Multiple choice

What is $\cot\left(\frac{\pi}{6}\right)$?

1. $\frac{\sqrt{3}}{3}$
2. $2$
3. $\sqrt{3}$
4. $\frac{1}{2}$

Question 91 · Multiple choice

Which value of $x$ is not in the domain of $y=\csc x$?

1. $x=\frac{\pi}{2}$
2. $x=\pi$
3. $x=\frac{\pi}{4}$
4. $x=\frac{3\pi}{2}$

Question 92 · Multiple choice

Which set is the range of the secant function, $y=\sec x$?

1. $[-1,1]$
2. $(-\infty,\infty)$
3. $(-1,1)$
4. $(-\infty,-1]\cup[1,\infty)$

Question 93 · Multiple choice

The cotangent graph $y=\cot x$ has a vertical asymptote along which of these lines?

1. $x=\frac{\pi}{2}$
2. $x=\frac{\pi}{4}$
3. $x=0$
4. $x=\frac{3\pi}{2}$

Question 94 · Multiple choice

If $\sec\theta=-2$ and $\frac{\pi}{2}<\theta<\pi$, what is $\theta$?

1. $\frac{4\pi}{3}$
2. $\frac{2\pi}{3}$
3. $\frac{\pi}{3}$
4. $\frac{5\pi}{6}$

Question 95 · Multiple choice

To three decimal places, what is $\sec(2.1)$?

1. $1.158$
2. $-0.505$
3. $-1.981$
4. $1.981$

Question 96 · Multiple choice

To three decimal places, what is $\cot(0.8)$?

1. $1.030$
2. $1.435$
3. $1.394$
4. $0.971$

Question 97 · Multiple choice

For $\cos x\ne1$, which expression is equivalent to $\dfrac{\sin^2x}{1-\cos x}$?

1. $1+\cos x$
2. $1-\cos x$
3. $\sin x$
4. $1+\sin x$

Question 98 · Multiple choice

Which expression is equivalent to $\sec^2x-1$?

1. $\tan^{2}x$
2. $\cot^{2}x$
3. $\sin^{2}x$
4. $\csc^{2}x$

Question 99 · Multiple choice

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

1. $\frac{\sqrt{6}-\sqrt{2}}{4}$
2. $\frac{\sqrt{3}}{4}$
3. $\frac{\sqrt{6}+\sqrt{2}}{4}$
4. $\frac{1+\sqrt{3}}{2}$

Question 100 · Multiple choice

If $\sin x=\frac{3}{5}$ and $x$ is in Quadrant II, what is $\sin(2x)$?

1. $\frac{24}{25}$
2. $\frac{6}{5}$
3. $-\frac{7}{25}$
4. $-\frac{24}{25}$

Question 101 · Multiple choice

Which expression is equivalent to $\cos(2x)$?

1. $2\sin^{2}x-1$
2. $1-2\sin^{2}x$
3. $1-\sin^{2}x$
4. $2\cos x$

Question 102 · Multiple choice

Which expression is equivalent to $\sin(3x)\cos x-\cos(3x)\sin x$?

1. $\sin(2x)$
2. $\sin(4x)$
3. $\cos(2x)$
4. $-\sin(2x)$

Question 103 · Multiple choice

What are the solutions of $\cos(2x)=\cos x$ on $[0,2\pi)$?

1. $\left\{0\right\}$
2. $\left\{0,\frac{2\pi}{3},\frac{4\pi}{3}\right\}$
3. $\left\{\frac{\pi}{3},\pi,\frac{5\pi}{3}\right\}$
4. $\left\{0,\pi\right\}$

Question 104 · Multiple choice

The expression $3\sin x-4\cos x$ can be written as $5\sin(x-\varphi)$ with $0<\varphi<\dfrac\pi2$. To three decimal places, what is $\varphi$?

1. $0.927$
2. $0.644$
3. $53.130$
4. $0.785$

Question 105 · Multiple choice

If $\cos\theta=0.31$ and $\dfrac{3\pi}{2}<\theta<2\pi$, what is $\sin(2\theta)$, to three decimal places?

1. $0.589$
2. $-0.589$
3. $0.620$
4. $-0.951$

Question 106 · Multiple choice

What are the rectangular coordinates of the point with polar coordinates $\left(6,\frac{\pi}{3}\right)$?

1. $\left(3\sqrt{3},3\right)$
2. $\left(6,\frac{\pi}{3}\right)$
3. $\left(3,3\sqrt{3}\right)$
4. $\left(-3,3\sqrt{3}\right)$

Question 107 · Multiple choice

Which polar coordinates $(r,\theta)$ with $r>0$ and $0\le\theta<2\pi$ describe the point $(0,-5)$?

1. $\left(5,\frac{\pi}{2}\right)$
2. $\left(-5,\frac{3\pi}{2}\right)$
3. $\left(5,\pi\right)$
4. $\left(5,\frac{3\pi}{2}\right)$

Question 108 · Multiple choice

Which polar coordinates describe the same point as $\left(2,\frac{\pi}{4}\right)$?

1. $\left(2,-\frac{\pi}{4}\right)$
2. $\left(-2,\frac{\pi}{4}\right)$
3. $\left(-2,\frac{5\pi}{4}\right)$
4. $\left(2,\frac{5\pi}{4}\right)$

Question 109 · Multiple choice

The polar equation $r=4\cos\theta$ is a circle. What are the rectangular coordinates of its center?

1. $\left(2,0\right)$
2. $\left(4,0\right)$
3. $\left(0,2\right)$
4. $\left(0,0\right)$

Question 110 · Multiple choice

Which is a polar form of the complex number $1+i\sqrt{3}$?

1. $2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$
2. $2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$
3. $4\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$
4. $\sqrt{2}\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$

Question 111 · Multiple choice

Which polar coordinates $(r,\theta)$ with $r>0$ and $0\le\theta<2\pi$ describe the point $(-1,-1)$?

1. $\left(\sqrt{2},\frac{\pi}{4}\right)$
2. $\left(2,\frac{5\pi}{4}\right)$
3. $\left(\sqrt{2},\frac{3\pi}{4}\right)$
4. $\left(\sqrt{2},\frac{5\pi}{4}\right)$

Question 112 · Multiple choice

What are the rectangular coordinates of the point with polar coordinates $\left(3,-\frac{\pi}{2}\right)$?

1. $\left(0,3\right)$
2. $\left(0,-3\right)$
3. $\left(-3,0\right)$
4. $\left(3,0\right)$

Question 113 · Multiple choice

Write the point $(-4,\,7)$ in polar coordinates $(r,\theta)$ with $r>0$ and $0\le\theta<2\pi$. What is $\theta$, to three decimal places?

1. $-1.052$
2. $1.052$
3. $2.090$
4. $5.232$

Question 114 · Multiple choice

To three decimal places, what is the $x$-coordinate of the point with polar coordinates $(4,\,2.5)$?

1. $2.394$
2. $3.996$
3. $3.205$
4. $-3.205$

Question 115 · Multiple choice

To three decimal places, what is the distance between the points with polar coordinates $(3,\,0.4)$ and $(5,\,2.1)$?

1. $6.153$
2. $2.000$
3. $5.831$
4. $7.618$

Question 116 · Multiple choice

The graph of a polar function $r=f(\theta)$ for $0\le\theta\le2\pi$ is shown. Which could be $f(\theta)$?

1. $2+2\cos\theta$
2. $2+2\sin\theta$
3. $2-2\cos\theta$
4. $1+2\cos\theta$

Question 117 · Multiple choice

How many petals does the rose curve $r=3\sin(2\theta)$ have?

1. $2$
2. $3$
3. $4$
4. $8$

Question 118 · Multiple choice

What is the greatest distance from the pole to a point on the curve $r=5\cos(3\theta)$?

1. $3$
2. $15$
3. $8$
4. $5$

Question 119 · Multiple choice

For which values of $\theta$ in $[0,2\pi)$ does the limaçon $r=1+2\cos\theta$ pass through the pole?

1. $\left\{\frac{\pi}{3},\frac{5\pi}{3}\right\}$
2. $\left\{\pi\right\}$
3. $\left\{\frac{\pi}{2},\frac{3\pi}{2}\right\}$
4. $\left\{\frac{2\pi}{3},\frac{4\pi}{3}\right\}$

Question 120 · Multiple choice

The graph of $r=f(\theta)=3\cos\theta$ is shown in the $\theta r$-plane for $0\le\theta\le2\pi$. For $\frac{\pi}{2}<\theta<\pi$, in which quadrant of the $xy$-plane are the points of the polar curve $r=f(\theta)$?

1. Quadrant IV
2. Quadrant II
3. Quadrant I
4. Quadrant III

Question 121 · Multiple choice

For which values of $\theta$ in $[0,\pi]$ is the curve $r=2\sin\theta$ at the pole?

1. $\left\{\frac{\pi}{2}\right\}$
2. $\left\{0\right\}$
3. $\left\{0,\pi\right\}$
4. $\left\{\pi\right\}$

Question 122 · Multiple choice

The polar curves $r=2\cos\theta$ and $r=1$ intersect at which values of $\theta$ in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$?

1. $\left\{-\frac{\pi}{6},\frac{\pi}{6}\right\}$
2. $\left\{-\frac{\pi}{3},\frac{\pi}{3}\right\}$
3. $\left\{\frac{\pi}{3}\right\}$
4. $\left\{0\right\}$

Question 123 · Multiple choice

The graph of $r=2+3\cos\theta$ passes through the pole once for $0\le\theta\le\pi$. To three decimal places, at what $\theta$?

1. $0.841$
2. $2.301$
3. $3.983$
4. $0.730$

Question 124 · Multiple choice

A spiral has polar equation $r=\dfrac{\theta}{2}$ for $0\le\theta\le2\pi$. To three decimal places, what is the $y$-coordinate of the point where $\theta=2$?

1. $-0.416$
2. $1.000$
3. $0.909$
4. $1.683$

Question 125 · Multiple choice

For the polar curve $r=3+2\sin\theta$, as $\theta$ increases on $\left(0,\frac{\pi}{2}\right)$, the distance from the pole is

1. decreasing, because $r$ is positive and decreasing
2. constant, because $r>0$
3. increasing, because $r$ is negative and decreasing
4. increasing, because $r$ is positive and increasing

Question 126 · Multiple choice

For $r=4\cos(2\theta)$, which describes the distance from the pole as $\theta$ increases on $\left(\frac{\pi}{4},\frac{\pi}{2}\right)$?

1. Increasing, because $r$ is negative and decreasing
2. Decreasing, because $r$ is decreasing
3. Decreasing, because $r$ is negative
4. Increasing, because $r$ is positive and increasing

Question 127 · Multiple choice

What is the average rate of change of $r=2+\sin\theta$ with respect to $\theta$ on $\left[0,\frac{\pi}{2}\right]$?

1. $\frac{\pi}{2}$
2. $1$
3. $\frac{2}{\pi}$
4. $2$

Question 128 · Multiple choice

For the polar function $r=e^{\theta/4}$, what is the average rate of change of $r$ with respect to $\theta$ on $\left[\frac{\pi}{2},\pi\right]$?

1. $0.712$
2. $0.453$
3. $1.396$
4. $0.227$

Question 129 · Multiple choice

For $r=1-2\cos\theta$, which describes the distance from the pole as $\theta$ increases on $\left(0,\frac{\pi}{3}\right)$?

1. Decreasing, because $r$ is negative and increasing
2. Increasing, because $r$ is increasing
3. Increasing, because $r$ is negative
4. Decreasing, because $r$ is positive and decreasing

Question 130 · Multiple choice

For the spiral $r=2\theta$, what is the average rate of change of $r$ with respect to $\theta$ on $[\pi,2\pi]$?

1. $2\pi$
2. $2$
3. $\pi$
4. $4$

Question 131 · Multiple choice

For $r=3\sin(2\theta)$, on which interval is the distance from the pole increasing as $\theta$ increases?

1. $\left(\frac{\pi}{4},\frac{\pi}{2}\right)$
2. $\left(\frac{3\pi}{4},\pi\right)$
3. $\left(0,\frac{\pi}{4}\right)$
4. $\left(\frac{\pi}{4},\frac{3\pi}{4}\right)$

Question 132 · Multiple choice

For the polar function $r=3\sin(2\theta)$, what is the average rate of change of $r$ with respect to $\theta$ on $[0.2,\,0.6]$, to three decimal places?

1. $1.628$
2. $2.745$
3. $2.035$
4. $4.070$

Question 133 · Multiple choice

For the polar function $r=2\theta+\sin\theta$, what is the average rate of change of $r$ with respect to $\theta$ on $[1,\,2.5]$, to three decimal places?

1. $1.838$
2. $2.757$
3. $2.000$
4. $4.220$

Question 134 · Multiple choice

For the polar function $r=1+2\cos\theta$, what is the average rate of change of $r$ with respect to $\theta$ on $[0.5,\,1.5]$, to three decimal places?

1. $1.614$
2. $-1.614$
3. $-0.807$
4. $-0.614$

Question 135 · Free response (6 points) · Modeling a periodic context

The tip of a windmill blade moves around a circle at a constant rate. Its height above the ground is $18$ meters at the lowest point and $42$ meters at the highest point. At time $t=0$ seconds the tip is at its highest point, and it makes one full turn every $6$ seconds. Let $h(t)$ be the height of the tip, in meters. Answer without a calculator.

1. (A(i)) Find the amplitude, the midline and the period of $h$. [1 point]
2. (A(ii)) Write an equation for $h(t)$. [1 point]
3. (B(i)) Find $h(1)$. [1 point]
4. (B(ii)) Find the first time $t>0$ at which the tip is $24$ meters above the ground. [1 point]
5. (C) On the interval $3<t<4.5$, is $h$ increasing or decreasing? Is the rate of change of $h$ increasing or decreasing? Give a reason for each answer. [2 points]

Question 136 · Free response (6 points) · Modeling a periodic context

The depth of the water at a pier rises and falls with the tides. The graph shows the depth $d$, in feet, $t$ hours after midnight. High tide, $9$ feet, is at $t=2$, and the next low tide, $3$ feet, is at $t=8$. Answer without a calculator.

1. (A(i)) Use the graph to find the midline, the amplitude and the period of $d$. [1 point]
2. (A(ii)) Write an equation for $d(t)$. [1 point]
3. (B(i)) Use your equation to find the depth at midnight. [1 point]
4. (B(ii)) Find the average rate of change of $d$ from $t=2$ to $t=5$, with units. [1 point]
5. (C) On which interval within $2<t<14$ is the depth decreasing and the graph of $d$ concave down? Give a reason. [2 points]

Question 137 · Free response (6 points) · Symbolic manipulation

Answer each part without a calculator. Show your work.

1. (A(i)) Find every $x$ with $0\le x<2\pi$ that satisfies $2\sin^2x+\sin x-1=0$. [1 point]
2. (A(ii)) Solve $\tan x=-\sqrt{3}$ for $0\le x<2\pi$. [1 point]
3. (B(i)) Rewrite $\dfrac{\cos(2\theta)}{\cos\theta-\sin\theta}$ as an expression with no fraction, for values of $\theta$ where $\cos\theta\ne\sin\theta$. [1 point]
4. (B(ii)) Find the exact value of $\sin\left(\frac{5\pi}{12}\right)$. [1 point]
5. (C) Find all $x$ in $[0,2\pi)$ for which $\cos x\ge\frac{1}{2}$. [2 points]

Question 138 · Free response (6 points) · Symbolic manipulation

Answer each part without a calculator. Show your work.

1. (A(i)) Convert the polar point $\left(4,\frac{5\pi}{6}\right)$ to rectangular coordinates. [1 point]
2. (A(ii)) Give polar coordinates $(r,\theta)$, with $r>0$ and $0\le\theta<2\pi$, for the point $(-3,3)$. [1 point]
3. (B(i)) For the polar function $r=2+4\cos\theta$, find $r$ at $\theta=\frac{2\pi}{3}$ and say what this tells you about the curve. [1 point]
4. (B(ii)) For $r=2+4\cos\theta$, is the distance between the point $(r,\theta)$ and the pole increasing or decreasing for $\frac{\pi}{2}<\theta<\frac{2\pi}{3}$? Give a reason. [1 point]
5. (C(i)) Find $\arcsin\left(\sin\frac{5\pi}{6}\right)$. [1 point]
6. (C(ii)) Find $\csc\left(\frac{7\pi}{6}\right)$. [1 point]