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

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

18 mins18 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

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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

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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

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

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6
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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

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7
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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

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8
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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.

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9
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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

10
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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.

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11
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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

12
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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)

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13
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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

14
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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

15
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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

16
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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

17
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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

18
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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