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

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

3 mins3 questions
1
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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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2
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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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3
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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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