AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsTrigonometric Functions

Trigonometric Functions (College Board AP® Precalculus): Exam Questions

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Trigonometric Functions

1
1 mark

The table gives selected values of the periodic function space f.

space t

0

2

4

6

8

10

12

space f open parentheses t close parentheses

5

8

5

2

5

8

5

Based on the values in the table, which of the following could be the period of space f?

  • 4

  • 6

  • 8

  • 10

Choose your answer

2
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1 mark

Which of the following angles has the same terminal ray in standard position as an angle of measure negative fraction numerator 3 pi over denominator 4 end fraction radians?

  • pi over 4

  • fraction numerator 3 pi over denominator 4 end fraction

  • fraction numerator 5 pi over denominator 4 end fraction

  • fraction numerator 7 pi over denominator 4 end fraction

Choose your answer

3
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1 mark

The graph of space y equals sin x for 0 less or equal than x less or equal than 2 pi is shown.

Graph of y = sin x over the interval 0 to 2 pi. The curve starts at the origin, increases to a maximum value of 1 at x = pi/2, returns to 0 at x = pi, decreases to a minimum value of -1 at x = 3 pi/2, and returns to 0 at x = 2 pi. The horizontal axis is labelled x with tick marks at pi/2, pi, 3 pi/2, and 2 pi. The vertical axis is labelled y with tick marks at -1 and 1.

Based on the graph, on which of the following intervals is \sin x decreasing?

  • open square brackets 0 comma pi over 2 close square brackets

  • open square brackets 0 comma pi close square brackets

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

  • open square brackets pi comma 2 pi close square brackets

Choose your answer

4
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1 mark

What is the exact value of sin open parentheses fraction numerator 11 pi over denominator 6 end fraction close parentheses?

  • negative fraction numerator square root of 3 over denominator 2 end fraction

  • negative 1 half

  • 1 half

  • fraction numerator square root of 3 over denominator 2 end fraction

Choose your answer

5
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1 mark

The function space f is given by space f open parentheses x close parentheses equals tan x.

For which of the following values of x does the graph of space f have a vertical asymptote?

  • 0

  • \dfrac{\pi}{4}

  • \dfrac{\pi}{2}

  • \pi

Choose your answer

6
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1 mark

The figure shows a circle centered at the origin with an angle of measure \theta radians in standard position. The terminal ray of the angle intersects the circle at point P, and point Q also lies on the circle. The coordinates of P are open parentheses x comma space y close parentheses, and Q is a reflection of P in the y-axis. Which of the following gives the x-coordinate of Q?

A circle centred at O on a Cartesian plane, showing point P on the circle in the second quadrant and Q in the circle in the first quadrant, with right-angled triangles formed by dashed lines. Angle θ marked at the centre from the x-axis to the ray to point P.

  • negative 4 cos theta

  • 4 cos theta

  • 4 sin theta

  • negative 4 sin theta

Choose your answer

7
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1 mark
Circle of radius 4 centred at the origin O. Point P is in Quadrant I on the circle, with angle θ shown from the positive x-axis to the ray OP. Dashed lines show the right triangle formed by P. Point Q is the reflection of P across the x-axis in Quadrant IV, with a dashed right triangle below the x-axis. The number 4 labels the radius on the positive x-axis.

The figure shows a circle of radius 4 centered at the origin O. The point P lies in Quadrant I on the circle, where the angle from the positive x-axis to the ray OP is \theta. The point Q is the reflection of P across the x-axis.

Which of the following gives the y-coordinate of Q?

  • -4\sin\theta

  • 4\sin\theta

  • 4\cos\theta

  • -4\cos\theta

Choose your answer

8
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1 mark

A circle is centered at the origin with radius 4. A point P starts at open parentheses 4 comma 0 close parentheses and travels a distance of 6 units along the circle in the counterclockwise direction. Which of the following gives the coordinates of P after this movement?

  • open parentheses 3.990 comma space 0.283 close parentheses

  • open parentheses 3.841 comma space minus 1.117 close parentheses

  • open parentheses 3.144 comma space 2.473 close parentheses

  • open parentheses 0.283 comma space 3.990 close parentheses

Choose your answer

9
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1 mark

The function g is given by g \left(x\right) = 2 \tan \left(\frac{x}{3}\right) + 1. What is the period of g?

  • \frac{\pi}{3}

  • \pi

  • 2\pi

  • 3\pi

Choose your answer

10
1 mark

The scatterplot shows values of a periodic function space f at integer values of t for 0 less or equal than t less or equal than 10.

Scatterplot showing selected values of a periodic function f for integer values of t from 0 to 10. Points are plotted at (0, 3), (1, 6), (2, 3), (3, 0), (4, 3), (5, 6), (6, 3), (7, 0), (8, 3), (9, 6), and (10, 3). The horizontal axis is labeled t and the vertical axis shows values from 0 to 6.

Based on the values shown, what is the period of space f?

  • 2

  • 4

  • 6

  • 8

Choose your answer

11
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1 mark

The function g is given by g open parentheses x close parentheses equals tan open parentheses 2 x minus pi over 3 close parentheses. At which of the following values of x for 0 less or equal than x less or equal than pi over 2 does the graph of g have a vertical asymptote?

  • \frac{\pi}{12}

  • \frac{\pi}{4}

  • \frac{5 \pi}{12}

  • \frac{\pi}{2}

Choose your answer

12
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1 mark

The terminal ray of an angle theta in standard position passes through the point P on the unit circle. If sin theta equals negative 1 half and theta is in the third quadrant, what is the value of \tan \theta?

  • - \sqrt{3}

  • - \dfrac{\sqrt{3}}{3}

  • \dfrac{\sqrt{3}}{3}

  • \sqrt{3}

Choose your answer

13
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1 mark

What is the exact value of sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses minus cos open parentheses fraction numerator 5 pi over denominator 6 end fraction close parentheses?

  • - \sqrt{3}

  • 0

  • \dfrac{\sqrt{3}}{2}

  • \sqrt{3}

Choose your answer

14
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1 mark

A point P on a circle of radius 6 centered at the origin begins at open parentheses 6 comma 0 close parentheses. P first moves clockwise along the circle through an arc of length 5 units, then moves counterclockwise along the circle through an arc of length 14 units. What is the space y-coordinate of the final position of P, expressed as a decimal approximation?

  • negative 4.441

  • 0.424

  • 4.339

  • 5.985

Choose your answer

15
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1 mark

The function space f is given by space f open parentheses x close parentheses equals cos x.

On which of the following intervals is space f open parentheses x close parentheses both negative and increasing?

  • open parentheses 2 pi comma space fraction numerator 5 pi over denominator 2 end fraction close parentheses

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

  • open parentheses 3 pi comma space fraction numerator 7 pi over denominator 2 end fraction close parentheses

  • open parentheses fraction numerator 7 pi over denominator 2 end fraction comma space 4 pi close parentheses

Choose your answer

16
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1 mark

The function g is periodic with a period of 4. The table gives values of g for selected values of x.

x

0

1

2

3

g open parentheses x close parentheses

3

5

1

- 1

What is the average rate of change of g on the interval open square brackets 57 comma 67 close square brackets?

  • - 6

  • - 3

  • - \dfrac{3}{5}

  • \dfrac{3}{5}

Choose your answer

17
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1 mark

Let space f be the function defined by space f open parentheses x close parentheses equals tan open parentheses pi over 2 x minus pi close parentheses. Let x subscript 1 and x subscript 2 be the two smallest positive values of x at which space f has a vertical asymptote. What is the value of x subscript 1 plus x subscript 2?

  • 2

  • 4

  • 6

  • 8

Choose your answer

18
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1 mark

The angle theta is in quadrant III and satisfies sin theta equals negative 2 over 3. What is the value of tan theta?

  • negative fraction numerator square root of 5 over denominator 2 end fraction

  • negative fraction numerator 2 square root of 5 over denominator 5 end fraction

  • \dfrac{2 \sqrt{5}}{5}

  • fraction numerator square root of 5 over denominator 2 end fraction

Choose your answer

19
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1 mark

An angle \theta in standard position has its terminal side passing through the point open parentheses negative 5 comma 12 close parentheses. What is the value of sin open parentheses theta plus pi close parentheses?

  • -\dfrac{12}{13}

  • -\dfrac{5}{13}

  • \dfrac{5}{13}

  • \dfrac{12}{13}

Choose your answer