AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsPolar Coordinates & Polar Functions

Polar Coordinates & Polar Functions (College Board AP® Precalculus): Exam Questions

32 mins32 questions
1
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1 mark

What are the rectangular coordinates of the point with polar coordinates open parentheses 2 comma pi over 3 close parentheses?

  • open parentheses 1 half comma fraction numerator square root of 3 over denominator 2 end fraction close parentheses

  • open parentheses fraction numerator square root of 3 over denominator 2 end fraction comma 1 half close parentheses

  • open parentheses 1 comma square root of 3 close parentheses

  • open parentheses square root of 3 comma 1 close parentheses

Choose your answer

2
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1 mark

What are the real part and the imaginary part of the complex number z equals negative 3 plus 5 i?

  • Real: negative 3, Imaginary: negative 5

  • Real: negative 3, Imaginary: 5

  • Real: negative 3, Imaginary: 5 i

  • Real: 5, Imaginary: negative 3

Choose your answer

3
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1 mark

The polar function space f is given by space f open parentheses theta close parentheses equals 4 sin theta plus 1. What is the average rate of change of r equals f open parentheses theta close parentheses with respect to theta on the interval open square brackets 0 comma pi over 2 close square brackets?

  • pi over 8

  • 4 over pi

  • 8 over pi

  • 4

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4
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1 mark

The table gives values for the polar function r equals f open parentheses theta close parentheses at selected values of theta. Assume that between any two consecutive values of theta listed in the table, space f is either always increasing or always decreasing.

\theta

0

pi over 4

pi over 2

fraction numerator 3 pi over denominator 4 end fraction

pi

fraction numerator 5 pi over denominator 4 end fraction

space f open parentheses theta close parentheses

4

5

4

1

2

3

Based on the table, at which value of theta does the graph of r equals f open parentheses theta close parentheses have a point that is relatively closest to the origin?

  • theta equals pi over 4

  • theta equals pi over 2

  • theta equals fraction numerator 3 pi over denominator 4 end fraction

  • theta equals pi

Choose your answer

5
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1 mark

What are the rectangular coordinates of the point with polar coordinates open parentheses 4 comma fraction numerator 5 pi over denominator 6 end fraction close parentheses?

  • open parentheses negative 2 square root of 3 comma space 2 close parentheses

  • open parentheses negative 2 comma space 2 square root of 3 close parentheses

  • open parentheses 2 square root of 3 comma space minus 2 close parentheses

  • open parentheses 2 square root of 3 comma space 2 close parentheses

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6
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1 mark

A complex number is given by z equals negative 3 plus 4 i and is represented by a point in the complex plane. What is the distance between this point and the origin?

  • 7

  • 5

  • 1

  • 25

Choose your answer

7
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1 mark

The table gives values for a polar function r equals f open parentheses theta close parentheses at selected values of theta.

space theta

0

pi over 4

pi over 2

space f open parentheses theta close parentheses

5

3

0

What is the average rate of change of r with respect to theta on the interval open square brackets 0 comma pi over 2 close square brackets?

  • negative 5

  • negative 10 over pi

  • negative 5 over pi

  • 10 over pi

Choose your answer

8
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1 mark

Which of the following polar equations represents a circle of radius 4 centered at the origin?

  • r equals 4

  • r equals 4 cos theta

  • r equals 4 plus cos theta

  • r equals cos open parentheses 4 theta close parentheses

Choose your answer

9
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1 mark

The polar function space f has values space f open parentheses pi over 6 close parentheses equals 2 and space f open parentheses pi over 2 close parentheses equals 8. Use the average rate of change of space f on the interval open square brackets pi over 6 comma pi over 2 close square brackets to estimate the value of space f open parentheses pi over 3 close parentheses.

  • 2

  • 3

  • 5

  • 8

Choose your answer

10
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1 mark

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates open parentheses negative 2 comma space 2 square root of 3 close parentheses. Which of the following is one way to express the complex number using its polar coordinates \left(r , \theta\right) ?

  • 2 square root of 3 cos open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses plus i open parentheses 2 square root of 3 sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses

  • 4 cos open parentheses pi over 3 close parentheses plus i open parentheses 4 sin open parentheses pi over 3 close parentheses close parentheses

  • 4 cos open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses plus i open parentheses 4 sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses

  • 4 cos open parentheses negative fraction numerator 2 pi over denominator 3 end fraction close parentheses plus i open parentheses 4 sin open parentheses negative fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses

Choose your answer

11
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1 mark

The figure shows the graph of the polar function r = f \left(\theta\right), where f \left(\theta\right) = 3 \sin \left(2 \theta\right), in the polar coordinate system for 0 \leq \theta \leq 2 \pi. There are five points labeled A, B, C, D, and E. If the domain of f is restricted to 0 \leq \theta \leq \pi, the portion of the given graph that remains consists of two pieces. One of those pieces is the portion of the graph in Quadrant I from B through A and back to B. Which of the following describes the other remaining piece?

A four-petaled rose curve r = 3sin(2 theta) in the polar coordinate system. The petals extend to r = 3. Point A is at the tip of the petal in Quadrant I (along the θ = π/4 terminal ray). Point B is at the origin. Point C is at the tip of the petal in Quadrant II (along the θ = 3π/4 terminal ray). Point D is at the tip of the petal in Quadrant III (along the θ = 5π/4 terminal ray). Point E is at the tip of the petal in Quadrant IV (along the θ = 7π/4 terminal ray). The petals are in Quadrants I, II, III, IV.

  • The portion of the graph in Quadrant II from B through C and back to B

  • The portion of the graph in Quadrant III from B through D and back to B

  • The portion of the graph in Quadrant IV from B through E and back to B

  • The portions of the graph in Quadrants II and III from B through C and back to B, and from B through D and back to B

Choose your answer

12
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1 mark

Consider the graph of the polar function r = f \left(\theta\right), where f \left(\theta\right) = 2 - 3 \cos \theta, in the polar coordinate system for 0 \leq \theta \leq 2 \pi. Which of the following statements is true about the distance between the point with polar coordinates open parentheses f open parentheses theta close parentheses comma space theta close parentheses and the origin?

  • The distance is decreasing for 0 \leq \theta \leq \arccos \left(\frac{2}{3}\right), because f \left(\theta\right) is negative and increasing on the interval.

  • The distance is increasing for \frac{\pi}{2} \leq \theta \leq \pi, because f \left(\theta\right) is negative and decreasing on the interval.

  • The distance is decreasing for 0 \leq \theta \leq \frac{\pi}{2}, because f \left(\theta\right) is positive and decreasing on the interval.

  • The distance is decreasing for \pi \leq \theta \leq \frac{3 \pi}{2}, because f \left(\theta\right) is negative and decreasing on the interval.

Choose your answer

13
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1 mark

Which of the following could be the graph of the polar curve with equation r = 3 + 2\sin\theta in the polar coordinate plane?

  • A polar curve in the polar coordinate plane with concentric circle reference rings at r = 1 through 5 and labelled values on the positive horizontal axis. The curve is a limaçon symmetric about the horizontal axis. It extends to r = 5 on the right side and has a pronounced concave dimple on the left side, with a minimum value of r = 1. The curve does not cross the origin and has no inner loop.
  • A polar curve in the polar coordinate plane with concentric circle reference rings at r = 1 through 5 and labelled values on the positive horizontal axis. The curve is a limaçon symmetric about the vertical axis. It extends to r = 5 at the top. A small inner loop is visible near the origin, with the loop located above the horizontal axis along the vertical axis. The outer loop is large and rounded, filling the upper half of the plane.
  • A polar curve in the polar coordinate plane with concentric circle reference rings at r = 1 through 5 and labelled values on the positive horizontal axis. The curve is a limaçon symmetric about the vertical axis. It extends to r = 5 at the top and has a shallow dimple at the bottom, with a minimum value of r = 1. The curve does not cross the origin and has no inner loop.
  • A polar curve in the polar coordinate plane with concentric circle reference rings at r = 1 through 5 and labelled values on the positive horizontal axis. The curve is a limaçon symmetric about the horizontal axis. It extends to r = 5 on the right side. A small inner loop is visible near the origin, with the loop located to the right of the origin along the positive horizontal axis. The outer loop is large and sweeps around the right half of the plane.

Choose your answer

14
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1 mark

A polar function is given by r = \sin\theta - 1. As \theta increases over the interval \left(\frac{\pi}{2}, \pi\right), which of the following describes the behavior of the distance from the pole and the position of the corresponding points relative to the polar axis?

  • The distance from the pole decreases, and the corresponding points are above the polar axis.

  • The distance from the pole decreases, and the corresponding points are below the polar axis.

  • The distance from the pole increases, and the corresponding points are above the polar axis.

  • The distance from the pole increases, and the corresponding points are below the polar axis.

Choose your answer

15
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1 mark

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates \left(-\sqrt{3}, 1\right). Which of the following is one way to express the complex number using its polar coordinates \left(r, \theta\right)?

  • \sqrt{3}\cos\left(\frac{5\pi}{6}\right) + i\left(\sqrt{3}\sin\left(\frac{5\pi}{6}\right)\right)

  • 2\cos\left(\frac{\pi}{6}\right) + i\left(2\sin\left(\frac{\pi}{6}\right)\right)

  • 2\cos\left(\frac{5\pi}{6}\right) + i\left(2\sin\left(\frac{5\pi}{6}\right)\right)

  • 2\cos\left(-\frac{5\pi}{6}\right) + i\left(2\sin\left(-\frac{5\pi}{6}\right)\right)

Choose your answer

16
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1 mark

A point P has polar coordinates open parentheses 3 comma pi over 4 close parentheses. Which of the following is also a polar representation of P?

  • \left(- 3 , - \dfrac{\pi}{4}\right)

  • \left(- 3 , \dfrac{\pi}{4}\right)

  • \left(- 3 , \dfrac{3 \pi}{4}\right)

  • \left(- 3 , \dfrac{5 \pi}{4}\right)

Choose your answer

17
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1 mark

A complex number z is given in polar form by z equals 4 cos open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses plus i open parentheses 4 sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses. What is the rectangular form a plus b i of z?

  • negative 2 minus 2 square root of 3 i

  • negative 2 plus 2 square root of 3 i

  • 2 plus 2 square root of 3 i

  • 2 square root of 3 minus 2 i

Choose your answer

18
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1 mark

The polar function space f is given by space f open parentheses theta close parentheses equals 2 sin theta plus 1. Which of the following is true about the graph of r equals f open parentheses theta close parentheses on the interval 0 less or equal than theta less or equal than 2 pi?

  • The minimum distance from the origin is 0, and the maximum distance from the origin is 1.

  • The minimum distance from the origin is 0, and the maximum distance from the origin is 3.

  • The minimum distance from the origin is 1, and the maximum distance from the origin is 1.

  • The minimum distance from the origin is 1, and the maximum distance from the origin is 3.

Choose your answer

19
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1 mark

The figure shows the graph of a polar function r = f \left(\theta\right) in the polar coordinate system for 0 \leq \theta \leq 2 \pi.

Polar graph showing a bold heart-shaped cardioid curve centred on the origin, symmetric about the vertical axis, with radial grid lines and labelled polar axis
Graph of r=f(θ)

Which of the following could be an expression for space f open parentheses theta close parentheses?

  • space f open parentheses theta close parentheses equals 2 minus 2 cos theta

  • space f open parentheses theta close parentheses equals 2 plus 2 cos theta

  • space f open parentheses theta close parentheses equals 2 minus 2 sin theta

  • space f open parentheses theta close parentheses equals 2 plus 2 sin theta

Choose your answer

20
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1 mark

The figure shows the graph of r equals f open parentheses theta close parentheses, where space f open parentheses theta close parentheses equals 1 plus cos theta, in the polar coordinate system for 0 less or equal than theta less or equal than 2 pi.

Polar coordinate grid with labelled axes and concentric circles, showing a bold cardioid-shaped curve symmetric about the horizontal polar axis.
Graph of r=f(θ)

On which of the following intervals does the graph appear in Quadrant IV?

  • \left(0 , \dfrac{\pi}{2}\right)

  • \left(\dfrac{\pi}{2} , \pi\right)

  • \left(\pi , \dfrac{3 \pi}{2}\right)

  • \left(\dfrac{3 \pi}{2} , 2 \pi\right)

Choose your answer

21
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1 mark

The figure shows the graph of r = f \left(\theta\right), where space f open parentheses theta close parentheses equals 1 plus 2 sin theta, in the polar coordinate system for 0 less or equal than theta less or equal than 2 pi.

Polar coordinate graph with concentric circles, showing a thick cardioid-like curve with a smaller inner loop, symmetric about the vertical axis
Graph of r=f(θ)

Which of the following statements is true about the distance between the point with polar coordinates open parentheses f open parentheses theta close parentheses comma theta close parentheses and the origin?

  • The distance is increasing for 0 less than theta less than pi over 2, because space f open parentheses theta close parentheses is positive and increasing on the interval.

  • The distance is increasing for pi over 2 less than theta less than pi, because space f open parentheses theta close parentheses is positive and increasing on the interval.

  • The distance is decreasing for fraction numerator 7 pi over denominator 6 end fraction less than theta less than fraction numerator 3 pi over denominator 2 end fraction, because space f open parentheses theta close parentheses is negative and decreasing on the interval.

  • The distance is decreasing for fraction numerator 3 pi over denominator 2 end fraction less than theta less than fraction numerator 11 pi over denominator 6 end fraction, because space f open parentheses theta close parentheses is positive and decreasing on the interval.

Choose your answer

22
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1 mark

The polar function space f is given by space f open parentheses theta close parentheses equals negative 2 plus sin theta for 0 \leq \theta \leq 2 \pi. At which value of theta does the graph of r equals f open parentheses theta close parentheses have a point that is relatively closest to the origin?

  • theta equals 0

  • theta equals pi over 2

  • theta equals pi

  • theta equals fraction numerator 3 pi over denominator 2 end fraction

Choose your answer

23
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1 mark

A point P has polar coordinates open parentheses negative 2 comma fraction numerator 5 pi over denominator 6 end fraction close parentheses. What are the rectangular coordinates of P?

  • open parentheses negative square root of 3 comma 1 close parentheses

  • open parentheses negative 1 comma square root of 3 close parentheses

  • open parentheses 1 comma negative square root of 3 close parentheses

  • open parentheses square root of 3 comma negative 1 close parentheses

Choose your answer

24
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1 mark

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates open parentheses negative 2 comma negative 2 close parentheses. Which of the following is one way to express the complex number using its polar coordinates open parentheses r comma theta close parentheses?

  • 2 cos open parentheses fraction numerator 5 pi over denominator 4 end fraction close parentheses plus i open parentheses 2 sin open parentheses fraction numerator 5 pi over denominator 4 end fraction close parentheses close parentheses

  • 2 square root of 2 cos open parentheses pi over 4 close parentheses plus i open parentheses 2 square root of 2 sin open parentheses pi over 4 close parentheses close parentheses

  • 2 square root of 2 cos open parentheses fraction numerator 5 pi over denominator 4 end fraction close parentheses plus i open parentheses 2 square root of 2 sin open parentheses fraction numerator 5 pi over denominator 4 end fraction close parentheses close parentheses

  • 2 square root of 2 cos open parentheses negative pi over 4 close parentheses plus i open parentheses 2 square root of 2 sin open parentheses negative pi over 4 close parentheses close parentheses

Choose your answer

25
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1 mark

Which of the following is the graph of the polar function r equals f open parentheses theta close parentheses, where space f open parentheses theta close parentheses equals 1 minus 2 cos theta, in the polar coordinate system for 0 less or equal than theta less or equal than 2 pi?

  • A polar curve in the polar coordinate plane with concentric circle reference rings labelled along the positive horizontal axis. The curve is a limaçon symmetric about the horizontal axis. It extends to r = 3 on the right side, reaching the point (3, 0). A small inner loop is visible near the origin on the right, with the loop located between the origin and the point (1, 0). The outer loop fills the right half of the plane.
  • A polar curve in the polar coordinate plane with concentric circle reference rings labelled along the positive horizontal axis. The curve is a limaçon symmetric about the vertical axis. It extends to r = 3 at the top, reaching the point (0, 3). A small inner loop is visible near the origin above the horizontal axis, with the loop located between the origin and the point (0, 1). The outer loop fills the upper half of the plane.
  • A polar curve in the polar coordinate plane with concentric circle reference rings labelled along the positive horizontal axis. The curve is a limaçon symmetric about the horizontal axis. It extends to r = 3 on the left side, reaching the point (-3, 0). A small inner loop is visible near the origin on the left, with the loop located between the origin and the point (-1, 0). The outer loop fills the left half of the plane.
  • A polar curve in the polar coordinate plane with concentric circle reference rings labelled along the positive horizontal axis. The curve is a limaçon symmetric about the vertical axis. It extends to r = 3 at the bottom, reaching the point (0, -3). A small inner loop is visible near the origin below the horizontal axis, with the loop located between the origin and the point (0, -1). The outer loop fills the lower half of the plane.

Choose your answer

26
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1 mark

Two complex numbers z subscript 1 equals square root of 7 plus 3 i and z subscript 2 equals negative 2 plus 3 i are represented by points in the complex plane. Which of the following correctly compares the distance from z subscript 1 to the origin and the distance from z subscript 2 to the origin?

  • The distance from z subscript 1 to the origin equals the distance from z subscript 2 to the origin.

  • The distance from z subscript 1 to the origin is less than the distance from z subscript 2 to the origin.

  • The distance from z subscript 1 to the origin is greater than the distance from z subscript 2 to the origin.

  • The relative distances cannot be determined without additional information.

Choose your answer

27
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1 mark

A polar function space f is given by f open parentheses theta close parentheses equals 1 plus 2 sin theta for 0 less or equal than theta less or equal than 2 pi. At which value of theta is the distance between the point with polar coordinates open parentheses f open parentheses theta close parentheses comma theta close parentheses and the origin the greatest?

  • theta equals 0

  • theta equals pi over 2

  • theta equals fraction numerator 7 pi over denominator 6 end fraction

  • theta equals fraction numerator 3 pi over denominator 2 end fraction

Choose your answer

28
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1 mark

Consider the graph of r equals f open parentheses theta close parentheses, where space f open parentheses theta close parentheses equals 2 sin theta minus 1, in the polar coordinate system for 0 less or equal than theta less or equal than 2 pi. On which of the following intervals does the value of r increase while the distance between the point with polar coordinates open parentheses f open parentheses theta close parentheses comma theta close parentheses and the origin decreases?

  • 0 less than theta less than pi over 6

  • pi over 6 less than theta less than pi over 2

  • pi over 2 less than theta less than fraction numerator 5 pi over denominator 6 end fraction

  • fraction numerator 5 pi over denominator 6 end fraction less than theta less than fraction numerator 3 pi over denominator 2 end fraction

Choose your answer

29
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1 mark

A polar function is given by r equals f open parentheses theta close parentheses, where space f open parentheses theta close parentheses equals 2 cos theta minus 1, in the polar coordinate system for 0 less or equal than theta less or equal than 2 pi. On which of the following intervals is the average rate of change of r with respect to theta equal to 0?

  • \left[0 , \frac{2 \pi}{3}\right]

  • \left[0 , \pi\right]

  • \left[\frac{\pi}{6} , \pi\right]

  • \left[\frac{\pi}{3} , \frac{5 \pi}{3}\right]

Choose your answer

30
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1 mark

A polar function r = f \left(\theta\right) is given by space f open parentheses theta close parentheses equals 4 sin open parentheses 3 theta close parentheses, in the polar coordinate system for 0 less or equal than theta less or equal than 2 pi. The graph is a rose curve. Which of the following describes the graph?

  • The graph consists of 6 petals, each of length 4.

  • The graph consists of 3 petals, each of length 4.

  • The graph consists of 6 petals, each of length 8.

  • The graph consists of 3 petals, each of length 8.

Choose your answer

31
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1 mark

The complex number z equals negative 2 plus 2 square root of 3   i is written in polar form as z equals open parentheses r cos theta close parentheses plus i open parentheses r sin theta close parentheses, where r greater than 0 and 0 less or equal than theta less than 2 pi. What is the value of theta?

  • \dfrac{7 \pi}{12}

  • \dfrac{2 \pi}{3}

  • \dfrac{3 \pi}{4}

  • \dfrac{5 \pi}{6}

Choose your answer

32
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1 mark

Consider the polar function r equals 1 minus 2 cos theta. On which of the following intervals is the distance between the graph of the function and the origin decreasing?

  • open square brackets 0 comma space pi over 3 close square brackets

  • open square brackets pi over 3 comma space pi over 2 close square brackets

  • open square brackets pi over 2 comma space pi close square brackets

  • open square brackets fraction numerator 5 pi over denominator 3 end fraction comma space 2 pi close square brackets

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