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 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

6
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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

7
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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

8
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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

9
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

10
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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

11
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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