AP PreCalculus Unit 3: Trigonometric and Polar Functions Practice Exam

Assessment for Unit 3: Trigonometric and Polar Functions

Estimated Time: 65 minutes

Exam Details

Questions: 49 total (45 multiple choice, 4 short answer)
Time: 65 minutes (recommended)
Topics Covered: Unit 3: Trigonometric and Polar Functions
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (45 Questions)

Select the one best answer for each question.

1. The depth of water at the end of a pier varies with the tides throughout the day. On a specific day, the high tide of 14 feet occurs at 2:00 a.m. The following low tide of 4 feet occurs at 8:00 a.m. The depth of the water can be modeled by a sinusoidal function $d(t)$, where $t$ is the time in hours since midnight. Which of the following best describes the amplitude and period of $d(t)$?

(A)Amplitude: 5 feet; Period: 6 hours(B)Amplitude: 5 feet; Period: 12 hours(C)Amplitude: 10 feet; Period: 6 hours(D)Amplitude: 10 feet; Period: 12 hours

2. A pendulum swings back and forth, and its horizontal displacement $x(t)$ from the center position is modeled by a periodic function, where $t$ is time in seconds. At $t=0$, the pendulum is at its farthest point to the right (positive displacement). It first reaches the center position at $t=0.5$ seconds and continues to its farthest point to the left. Which of the following correctly identifies the period of the function and the behavior of the graph of $x(t)$ on the interval $(0, 1)$?

(A)Period: 2 seconds; The graph is decreasing on $(0, 1)$.(B)Period: 2 seconds; The graph is decreasing on $(0, 0.5)$ and increasing on $(0.5, 1)$.(C)Period: 4 seconds; The graph is decreasing on $(0, 1)$.(D)Period: 4 seconds; The graph is decreasing on $(0, 0.5)$ and increasing on $(0.5, 1)$.

3. The function $h(t) = 10 + 4\sin\left(\frac{\pi}{4}t\right)$ models the height of a liquid in a container, in inches, at time $t$ seconds, where $0 \le t \le 8$. For which interval of $t$ is the height of the liquid strictly less than 8 inches?

(A)$\left$($\frac{14}{3}$, $\frac{22}{3}\right$)(B)$\left$($\frac{7}{3}$, $\frac{11}{3}\right$)(C)$\left[0, \frac{2}{3}\right) \cup \left(\frac{10}{3}, 8\right]$(D)$\left$($\frac{14}{3}$, 8$\right$]

4. Which of the following represents all solutions to the equation $2\cos(3x) + \sqrt{2} = 0$ ?

(A)x = $\frac{3\pi}{4} $+ 2$\pi $n $\text{ and } $x = $\frac{5\pi}{4} $+ 2$\pi $n(B)x = $\frac{\pi}{4} $+ $\frac{2\pi n}{3} \text{ and } $x = $\frac{5\pi}{12} $+ $\frac{2\pi n}{3}$(C)x = $\frac{\pi}{4} $+ 2$\pi $n $\text{ and } $x = $\frac{7\pi}{12} $+ 2$\pi $n(D)x = $\pm \frac{\pi}{12} $+ $\frac{2\pi n}{3}$

5. Which of the following expressions represents all real solutions for $x$ in the equation $4\sin(x) + 3 = 0$ ?

(A)x = $\arcsin\left$(-$\frac{3}{4}\right$) + 2$\pi $n(B)x = $\pm \arcsin\left$(-$\frac{3}{4}\right$) + 2$\pi $n(C)x = $\arcsin\left$(-$\frac{3}{4}\right$) + 2$\pi $n $\text{ and } $x = $\pi $- $\arcsin\left$(-$\frac{3}{4}\right$) + 2$\pi $n(D)x = $\arcsin\left$($\frac{3}{4}\right$) + $\pi $n

6. [Skill: 2.B | Topic: 3.11] The function $f$ is given by $f(\theta) = \csc(2\theta)$. Which of the following describes the values of $\theta$ for which the graph of $f$ has vertical asymptotes?

(A) $\theta $= n$\pi $, where n is an integer(B) $\theta $= $\frac{\pi}{2} $+ n$\pi $, where n is an integer(C) $\theta $= $\frac{n\pi}{2} $, where n is an integer(D) $\theta $= $\frac{\pi}{4} $+ $\frac{n\pi}{2} $, where n is an integer

7. [Skill: 3.A | Topic: 3.11] Let $g$ be the function defined by $g(x) = 3 \sec x - 2$. Which of the following is the range of $g$?

(A) [-5, 1] (B) (-$\infty$, -5] $\cup $[1, $\infty$) (C) (-$\infty$, -1] $\cup $[5, $\infty$) (D) (-5, 1)

8. [Skill: 3.B | Topic: 3.11] The function $h$ is defined by $h(x) = \sec x$. On which of the following intervals is the graph of $h$ both negative and increasing?

(A) (0, $\frac{\pi}{2}$) (B) ($\frac{\pi}{2}$, $\pi$) (C) ($\pi$, $\frac{3\pi}{2}$) (D) ($\frac{3\pi}{2}$, 2$\pi$)

9. The function $f$ is defined by $f(\theta) = \frac{\sec^2 \theta - 1}{\sin^2 \theta}$ for all values of $\theta$ where the expression is defined. Which of the following is an equivalent expression for $f(\theta)$?

(A)$\sec^2 \theta$(B)$\csc^2 \theta$(C)$\tan^2 \theta$(D)1

10. Which of the following expressions is equivalent to $\sin(3x)\cos(x) - \cos(3x)\sin(x)$ ?

(A)$\sin$(4x)(B)$\sin$(2x)(C)$\cos$(2x)(D)$\cos$(4x)

11. What are all values of $\theta$ in the interval $0 \le \theta < 2\pi$ that satisfy the equation $2\cos^2 \theta + \sin \theta = 1$?

(A)$\frac{\pi}{6}$, $\frac{5\pi}{6}$, $\frac{3\pi}{2}$(B)$\frac{\pi}{2}$, $\frac{7\pi}{6}$, $\frac{11\pi}{6}$(C)$\frac{\pi}{6}$, $\frac{5\pi}{6}$, $\frac{\pi}{2}$(D)$\frac{7\pi}{6}$, $\frac{11\pi}{6}$, $\frac{3\pi}{2}$

12. A point in the rectangular coordinate system is given by $(-2, -2\sqrt{3})$. Which of the following represents the polar coordinates $(r, \theta)$ of this point?

(A)$(4, \frac{2\pi}{3})$(B)$(4, \frac{5\pi}{3})$(C)$(-4, \frac{\pi}{3})$(D)$(-4, \frac{4\pi}{3})$

13. Which of the following gives the rectangular coordinates of the point with polar coordinates $(-6, \frac{5\pi}{4})$?

(A)$(-3\sqrt{2}, -3\sqrt{2})$(B)$(3\sqrt{2}, 3\sqrt{2})$(C)$(-3\sqrt{2}, 3\sqrt{2})$(D)$(3\sqrt{2}, -3\sqrt{2})$

14. A complex number $z$ is defined by $z = 8(\cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4})$. Which of the following represents $z$ in the form $a + bi$?

(A)$4\sqrt{2} - 4i\sqrt{2}$(B)$-4\sqrt{2} + 4i\sqrt{2}$(C)$4\sqrt{2} + 4i\sqrt{2}$(D)$-4\sqrt{2} - 4i\sqrt{2}$

15. The polar function $r = f(\theta)$ is defined on the interval $0 \le \theta \le \pi$. Given that $f(\theta) < 0$ for all $\theta$ in the interval $\frac{\pi}{2} < \theta < \pi$, in which quadrant of the Cartesian plane will the graph of $r = f(\theta)$ appear corresponding to this interval?

(A)Quadrant I(B)Quadrant II(C)Quadrant III(D)Quadrant IV

Refer to the figure below.

16. The graph of the polar function $r = 1 + 2\cos(\theta)$ is shown in the polar coordinate system. The graph contains a larger outer loop and a smaller inner loop that passes through the pole. Which of the following intervals for $\theta$ corresponds to the portion of the graph that forms the inner loop?

(A)$\frac{\pi}{3} < \theta < \frac{5\pi}{3}$(B)$\frac{2\pi}{3} < \theta < \frac{4\pi}{3}$(C)$0 < \theta < \frac{2\pi}{3}$(D)$\frac{4\pi}{3} < \theta < 2\pi$

17. A polar function is given by $r = f(\theta)$. On the interval $\frac{\pi}{2} < \theta < \pi$, the function $f$ is negative and increasing. Which of the following best describes the motion of a point on the graph of $r = f(\theta)$ as $\theta$ increases from $\frac{\pi}{2}$ to $\pi$?

(A)The point moves through the second quadrant and its distance from the origin increases.(B)The point moves through the second quadrant and its distance from the origin decreases.(C)The point moves through the fourth quadrant and its distance from the origin increases.(D)The point moves through the fourth quadrant and its distance from the origin decreases.

18. Let $r = f(\theta)$ be a polar function where $f(\theta) < 0$ for all $\theta$ in the interval $\frac{\pi}{2} < \theta < \pi$. If the rate of change of $r$ with respect to $\theta$ is positive on this interval, which of the following statements is true about the distance between the point on the curve and the origin?

(A)The distance is increasing because $r$ is increasing and negative.(B)The distance is decreasing because $r$ is increasing and negative.(C)The distance is increasing because the rate of change of $r$ is positive.(D)The distance is decreasing because the rate of change of $r$ is positive.

19. A polar function is defined by $r = 4 - 2\sin(\theta)$. What is the average rate of change of $r$ with respect to $\theta$ on the interval $0 \le \theta \le \frac{\pi}{2}$?

(A)-$\frac{4}{\pi}$(B)-$\frac{2}{\pi}$(C)$\frac{4}{\pi}$(D)-2

20. The graph of the polar function $r = f(\theta)$ is analyzed on the interval $\pi < \theta < 2\pi$. It is known that $f(\theta) < 0$ on this interval. If $f'(\theta)$ changes from negative to positive at $\theta = \frac{3\pi}{2}$, which of the following describes the point on the graph at $\theta = \frac{3\pi}{2}$?

(A)The point is at a relative minimum distance from the pole.(B)The point is at a relative maximum distance from the pole.(C)The point is at the pole.(D)The curve has a vertical tangent line at this point.

21. An angle $\theta$ is in standard position in the $xy$-plane. The terminal ray of the angle intersects the unit circle at point $P$. If the measure of angle $\theta$ is $\frac{2\pi}{3}$ radians, what are the coordinates of point $P$?

(A)$\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$(B)$\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$(C)$\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$(D)$\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$

22. The terminal ray of an angle $\theta$ in standard position intersects the unit circle at the point $P\left( -\frac{5}{13}, -\frac{12}{13} \right)$. What is the value of $\tan \theta$?

(A)$\frac{5}{12}$(B)$\frac{12}{5}$(C)$-\frac{5}{12}$(D)$-\frac{12}{5}$

23. For an angle $\theta$ in standard position, the terminal ray intersects the unit circle at a point $(x, y)$. If $\tan \theta = -2$ and $x > 0$, what is the value of $\sin \theta$?

(A)$\frac{1}{\sqrt{5}}$(B)$-\frac{1}{\sqrt{5}}$(C)$\frac{2}{\sqrt{5}}$(D)$-\frac{2}{\sqrt{5}}$

24. The terminal ray of an angle $\theta$ in standard position intersects a circle centered at the origin at the point $P(-8, 15)$. What is the value of $\cos \theta$?

(A)$-\frac{8}{17}$(B)$-\frac{8}{15}$(C)$\frac{15}{17}$(D)$-\frac{15}{8}$

25. An angle $\theta$ is in standard position, and its terminal ray intersects a circle of radius $6$ centered at the origin at point $Q$. If $\theta = \frac{5\pi}{6}$, what are the coordinates of point $Q$?

(A)$(-3, 3\sqrt{3})$(B)$(-3\sqrt{3}, 3)$(C)$(3\sqrt{3}, -3)$(D)$(-\frac{\sqrt{3}}{2}, \frac{1}{2})$

26. Circle $A$ and Circle $B$ are both centered at the origin. The radius of Circle $B$ is $k$ times the radius of Circle $A$, where $k > 1$. The terminal ray of an angle $\theta$ in standard position intersects Circle $A$ at point $(x_A, y_A)$ and Circle $B$ at point $(x_B, y_B)$. Which of the following correctly expresses $y_B$ in terms of $y_A$?

(A)$y_B = y_A + k$(B)$y_B = \frac{y_A}{k}$(C)$y_B = k \cdot y_A$(D)$y_B = y_A$

27. A point $P$ moves counterclockwise along the unit circle defined by $x^2 + y^2 = 1$, starting at the point $(1, 0)$. The angle of rotation, in radians, is given by $\theta$. Which of the following functions $f(\theta)$ represents the horizontal displacement of point $P$ from the $y$-axis?

(A)$f(\theta) = \sin \theta$, because the horizontal displacement corresponds to the $y$-coordinate on the unit circle.(B)$f(\theta) = \sin \theta$, because the horizontal displacement corresponds to the $x$-coordinate on the unit circle.(C)$f(\theta) = \cos \theta$, because the horizontal displacement corresponds to the $y$-coordinate on the unit circle.(D)$f(\theta) = \cos \theta$, because the horizontal displacement corresponds to the $x$-coordinate on the unit circle.

28. Consider the function $g(\theta) = \sin \theta$, which represents the vertical displacement of a point on the unit circle. On which of the following intervals is the graph of $g(\theta)$ both negative and decreasing?

(A)$(0, \frac{\pi}{2})$(B)$(\frac{\pi}{2}, \pi)$(C)$(\pi, \frac{3\pi}{2})$(D)$(\frac{3\pi}{2}, 2\pi)$

29. The function $f$ is given by $f(\theta) = -4 \cos\left(3\theta - \frac{\pi}{2}\right) + 1$. What are the amplitude and period of $f$?

(A)Amplitude: $4$; Period: $\frac{2\pi}{3}$(B)Amplitude: $-4$; Period: $\frac{2\pi}{3}$(C)Amplitude: $4$; Period: $6\pi$(D)Amplitude: $1$; Period: $3$

30. A sinusoidal function $g$ has a minimum value of $-5$ and a maximum value of $9$. Which of the following gives the values of the amplitude and the midline of the graph of $g$?

(A)Amplitude: $2$; Midline: $y = 7$(B)Amplitude: $7$; Midline: $y = 2$(C)Amplitude: $7$; Midline: $y = 7$(D)Amplitude: $14$; Midline: $y = 2$

31. The vertical position of a point on a vibrating string is modeled by the function $p(t) = \sin(8\pi t)$, where $t$ is measured in seconds. What is the frequency of the vibration in cycles per second?

(A)$\frac{1}{4}$(B)$4$(C)$8$(D)$8\pi$

32. [Skill: 2.B | Topic: 3.6] A sinusoidal function is given by the equation $f(\theta) = -4 \sin\left(\frac{\theta}{3}\right) + 2$. Which of the following correctly identifies the amplitude and period of the function?

(A)Amplitude: 4, Period: $6\pi$(B)Amplitude: -4, Period: $6\pi$(C)Amplitude: 4, Period: $\frac{2\pi}{3}$(D)Amplitude: 4, Period: $3$

33. [Skill: 1.C | Topic: 3.6] The graph of the function $h$ is the result of transforming the graph of $f(x) = \sin(x)$. The transformations applied to $f$ are a vertical dilation by a factor of 5, a horizontal compression by a factor of $\frac{1}{2}$, and a horizontal translation of $\frac{\pi}{4}$ units to the right. Which of the following defines $h(x)$?

(A) $h(x) = 5 \sin\left(2\left(x - \frac{\pi}{4}\right)\right)$(B) $h(x) = 5 \sin\left(2x - \frac{\pi}{4}\right)$(C) $h(x) = \frac{1}{5} \sin\left(\frac{1}{2}\left(x + \frac{\pi}{4}\right)\right)$(D) $h(x) = 5 \sin\left(\frac{1}{2}\left(x - \frac{\pi}{4}\right)\right)$

34. [Skill: 1.B | Topic: 3.7] A scientist records the vertical position y , in centimeters, of a mass suspended from a spring at various times t , in seconds. The data is shown in the table below. | t | 0 | 2 | 4 | 6 | 8 | |:---:|:---:|:---:|:---:|:---:|:---:| | y | 12 | 18 | 12 | 6 | 12 | Which of the following functions best models the data?

(A)y = 6 $\sin$($\frac{\pi}{4}$t) + 12(B)y = 6 $\cos$($\frac{\pi}{4}$t) + 12(C)y = 12 $\sin$($\frac{\pi}{4}$t) + 6(D)y = 6 $\sin$($\frac{\pi}{2}$t) + 12

35. [Skill: 2.B | Topic: 3.7] The number of hours of daylight in a certain city varies sinusoidally over the course of a year. The maximum number of daylight hours is 14.5, occurring at t = 172 days. The minimum number of daylight hours is 9.5, occurring at t = 355 days. Which of the following functions d(t) could model the number of daylight hours t days after the start of the year?

(A)d(t) = 2.5 $\cos$($\frac{2\pi}{366}$(t - 172)) + 12(B)d(t) = 5 $\cos$($\frac{2\pi}{366}$(t + 172)) + 9.5(C)d(t) = 2.5 $\cos$($\frac{2\pi}{183}$(t - 172)) + 12(D)d(t) = 2.5 $\sin$($\frac{2\pi}{366}$(t - 172)) + 12

36. [Skill: 2.B | Topic: 3.7] A Ferris wheel with a diameter of 50 feet rotates at a constant speed, completing one full revolution every 3 minutes. A rider boards the wheel at its lowest point, which is 5 feet above the ground, at time t = 0 minutes. Which of the following functions models the rider's height h(t) , in feet, above the ground?

(A)h(t) = -25 $\cos$($\frac{2\pi}{3}$t) + 30(B)h(t) = -25 $\cos$($\frac{2\pi}{3}$t) + 5(C)h(t) = 25 $\cos$($\frac{2\pi}{3}$t) + 30(D)h(t) = -50 $\cos$(3t) + 30

Refer to the figure below.

37. The figure below displays a unit circle in the xy-plane. A ray extending from the origin forms an angle θ in standard position and intersects the unit circle at point P with coordinates (x, y). Which of the following expressions represents the slope of the terminal ray and is equivalent to tan(θ)?

(A)y(B)x(C)y / x(D)x / y

38. The function f is defined by f(θ) = tan(θ). Which of the following statements correctly describes the domain of f and the behavior of its graph at the excluded values?

(A)The domain is all real numbers except θ = kπ for integers k. The graph has vertical asymptotes at these values.(B)The domain is all real numbers except θ = π/2 + kπ for integers k. The graph has vertical asymptotes at these values.(C)The domain is all real numbers except θ = kπ for integers k. The graph has holes (removable discontinuities) at these values.(D)The domain is all real numbers except θ = π/2 + kπ for integers k. The graph has holes (removable discontinuities) at these values.

39. The function g is given by g(x) = 3tan(2x - π). What are the period of g and the equations of the vertical asymptotes of the graph of g?

(A)Period: π; Vertical Asymptotes: x = π/4 + kπ/2, where k is an integer.(B)Period: π/2; Vertical Asymptotes: x = π/2 + kπ, where k is an integer.(C)Period: π; Vertical Asymptotes: x = π/2 + kπ, where k is an integer.(D)Period: π/2; Vertical Asymptotes: x = 3π/4 + kπ/2, where k is an integer.

40. The function $f$ is defined by $f(x) = \arcsin(x)$. Which of the following tables correctly describes the domain and range of $f$?

(A)Domain: $[-\frac{\pi}{2}, \frac{\pi}{2}]$; Range: $[-1, 1]$(B)Domain: $[-1, 1]$; Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$(C)Domain: $[0, \pi]$; Range: $[-1, 1]$(D)Domain: $[-1, 1]$; Range: $[0, \pi]$

41. What is the value of $\arccos\left(-\frac{\sqrt{2}}{2}\right)$ ?

(A)-$\frac{\pi}{4}$(B)$\frac{\pi}{4}$(C)$\frac{3\pi}{4}$(D)$\frac{5\pi}{4}$

42. Which of the following is equivalent to $\arctan\left(\tan\left(\frac{4\pi}{3}\right)\right)$ ?

(A)$\frac{\pi}{3}$(B)-$\frac{\pi}{3}$(C)$\frac{4\pi}{3}$(D)-$\frac{2\pi}{3}$

43. [Skill: 2.B | Topic: 3.4] A point $P$ moves counterclockwise along the unit circle in the $xy$-plane. The coordinates of $P$ are given by $(x, y) = (\cos \theta, \sin \theta)$, where $\theta$ is the angle in standard position measured in radians. As the angle $\theta$ increases from $\frac{\pi}{2}$ to $\pi$, which of the following statements correctly describes the behavior of the sine and cosine functions?

(A)The values of $\sin \theta$ decrease from 1 to 0, and the values of $\cos \theta$ decrease from 0 to -1.(B)The values of $\sin \theta$ increase from 0 to 1, and the values of $\cos \theta$ decrease from 0 to -1.(C)The values of $\sin \theta$ decrease from 1 to 0, and the values of $\cos \theta$ increase from -1 to 0.(D)The values of $\sin \theta$ increase from -1 to 0, and the values of $\cos \theta$ decrease from 1 to 0.

44. [Skill: 2A | Topic: 3.1] The water level in a harbor changes periodically due to the tides. At 2:00 AM ($t = 2$ hours), the water level reaches its maximum height of 15 meters. At 8:00 AM ($t = 8$ hours), the water level reaches its minimum height of 7 meters. If the water level $H(t)$ is modeled as a periodic function of time $t$ in hours, which of the following statements is true?

(A)The period of the function is 6 hours and the midline is at $H = 11$ meters.(B)The period of the function is 12 hours and the amplitude is 4 meters.(C)The period of the function is 12 hours and the midline is at $H = 4$ meters.(D)The period of the function is 6 hours and the amplitude is 8 meters.

45. [Skill: 2B | Topic: 3.6] A sinusoidal function $f(\theta)$ has the following key points over one cycle: - It crosses its midline at $\left(\frac{\pi}{4}, 1\right)$ and is increasing there. - It reaches a maximum at $\left(\frac{\pi}{2}, 4\right)$. - It crosses its midline again at $\left(\frac{3\pi}{4}, 1\right)$. - It reaches a minimum at $(\pi, -2)$. - It crosses its midline again at $\left(\frac{5\pi}{4}, 1\right)$. Which of the following could be an equation for $f(\theta)$?

(A)$f(\theta)=3\sin\left(2\left(\theta-\frac{\pi}{4}\right)\right)+1$(B)$f(\theta)=3\sin\left(2\left(\theta+\frac{\pi}{4}\right)\right)+1$(C)$f(\theta)=3\sin\left(\frac{1}{2}\left(\theta-\frac{\pi}{4}\right)\right)+1$(D)$f(\theta)=-3\sin\left(2\left(\theta-\frac{\pi}{4}\right)\right)+1$

B. Short Answer Questions (4 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 46:

Question 47:

Question 48:

Question 49:

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