AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsTrigonometric Functions

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

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Trigonometric Functions

1
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3 marks

The graph of space f open parentheses x close parentheses equals cos x is shown for 0 \leq x \leq 4 \pi.

Graph of y = cos x over the interval 0 to 4 pi. The curve starts at (0, 1), decreases through the x-axis to a minimum of -1 at x = pi, increases back through the x-axis to a maximum of 1 at x = 2 pi, decreases to -1 at x = 3 pi, and increases back to 1 at x = 4 pi, completing two full periods. The horizontal axis is labelled x with tick marks at pi, 2 pi, 3 pi, and 4 pi. The vertical axis is labelled y with tick marks at -1 and 1.

(i) What is the period of space f?

(ii) What are all values of x in open square brackets 0 comma 4 pi close square brackets at which space f attains its minimum value?

(iii) What are all intervals of x contained in open square brackets 0 comma 4 pi close square brackets on which space f is increasing?

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2
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3 marks

The angle theta equals fraction numerator 2 pi over denominator 3 end fraction is in standard position.

(i) Find the exact value of sin theta by using an appropriate reference angle in the interval 0 less or equal than theta less or equal than pi over 2. Show the work that leads to your answer.

(ii) Find the exact value of cos theta by using an appropriate reference angle in the interval 0 less or equal than theta less or equal than pi over 2. Show the work that leads to your answer.

(iii) Find the exact value of tan theta.

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3
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2 marks

Consider the tangent function space f given by space f open parentheses theta close parentheses equals tan open parentheses theta close parentheses on the open interval open parentheses 0 comma pi over 2 close parentheses.

(i) Explain why tan open parentheses theta close parentheses greater than 0 for all theta in the interval open parentheses 0 comma pi over 2 close parentheses.

(ii) Is space f strictly increasing or strictly decreasing on open parentheses 0 comma pi over 2 close parentheses? Give a reason for your answer using the definition of tangent in terms of sine and cosine.

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4
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2 marks

A circle is centered at the origin with radius 4. The point P equals open parentheses negative 2 square root of 3 comma 2 close parentheses lies on the circle at an angle of fraction numerator 5 pi over denominator 6 end fraction radians measured counterclockwise from the positive x-axis.

(i) Determine the exact value of sin open parentheses fraction numerator 5 pi over denominator 6 end fraction close parentheses. Show the computations that lead to your answer.

(ii) Determine the exact value of cos open parentheses fraction numerator 5 pi over denominator 6 end fraction close parentheses. Show the computations that lead to your answer.

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5
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2 marks

The point P equals open parentheses negative 3 comma 4 close parentheses lies on a circle centered at the origin. The angle theta in standard position has its terminal ray passing through P.

(i) Explain why tan open parentheses theta close parentheses is equal to the slope of the terminal ray of theta t for any angle in standard position.

(ii) Use the result from part (i) to find the exact value of tan open parentheses theta close parentheses. Show the computations that lead to your answer.

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6
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3 marks

The function space f is periodic with period 7. Selected values of space f are given in the table below.

space x

0

1

2

3

4

5

6

space f open parentheses x close parentheses

5

2

-3

8

1

6

-4

(i) Find space f open parentheses 23 close parentheses. Show the computations that lead to your answer.

(ii) Find space f open parentheses negative 11 close parentheses. Show the computations that lead to your answer.

(iii) Find all values of x in the interval open square brackets 0 comma 21 close square brackets for which space f open parentheses x close parentheses equals 6.

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7
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2 marks

The figure shows a unit circle centered at the origin O in the x y-plane. The point P is on the unit circle, and theta is the angle between the positive x-axis and the radius O P.

Unit circle diagram with centre O at the origin. A point P is shown in the first quadrant on the circle. The angle from the positive x-axis to the radius OP is labelled theta. A vertical dashed line from P to the x-axis is labelled sin theta. A horizontal dashed line from P to the y-axis is labelled cos theta. Tick marks at plus and minus 1 are shown on both axes.

(i) Using the figure, explain what sin open parentheses theta close parentheses represents geometrically on the unit circle.

(ii) Hence explain why negative 1 less or equal than sin open parentheses theta close parentheses less or equal than 1 for all values of \theta.

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8a
1 mark

A point P moves counterclockwise around a circular track of radius 8 meters centered at the origin. At time t equals 0 seconds, P is at the position open parentheses 8 comma 0 close parentheses. P moves along the track at a constant rate of 5 meters per second.

The angle theta in standard position has its terminal ray passing through P. Find the value of theta, in radians, at time t equals 4 seconds. Show the work that leads to your answer.

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8b
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2 marks

Find the coordinates of P at time t equals 4 seconds. Express the coordinates as decimal approximations. Show the work that leads to your answer.

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9a
1 mark

The function h is periodic with a period of 6. The table gives values for h open parentheses x close parentheses at selected values of x within one period.

x

0

1

2

3

4

5

h open parentheses x close parentheses

1

4

9

4

1

- 3

Find the value of h open parentheses 50 close parentheses. Show the work that leads to your answer.

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9b
1 mark

Find the average rate of change of h on the interval open square brackets 20 comma 23 close square brackets. Show the work that leads to your answer.

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10
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3 marks

The function g is given by g open parentheses x close parentheses equals negative 3 tan open parentheses x over 4 close parentheses.

(i) Determine the period of g. Show the computations that lead to your answer.

(ii) State whether g is strictly increasing or strictly decreasing on any open interval between consecutive vertical asymptotes. Explain why, using the transformations applied to the parent tangent function tan x.

(iii) Explain why the parent tangent function tan x has vertical asymptotes, referencing the sine and cosine functions.

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11a
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1 mark

The function g is given by g open parentheses x close parentheses equals a tan open parentheses b x close parentheses plus d, where a, b, and d are constants with a greater than 0 and b greater than 0. The graph of g:

  • has consecutive vertical asymptotes at x equals negative pi over 6 and x equals pi over 6

  • passes through the points open parentheses 0 comma 1 close parentheses and open parentheses pi over 12 comma 7 close parentheses

Determine the value of b. Show the work that leads to your answer.

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11b
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1 mark

Determine the value of d. Show the work that leads to your answer.

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11c
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1 mark

Determine the value of a. Show the work that leads to your answer.

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12a
1 mark
Graph of a periodic function f shown for 0 less than or equal to x less than or equal to 7. The function has period 3, with relative minima at (0, 2), (3, 2) and (6, 2), and relative maxima at (1, 5), (4, 5) and (7, 5). On each interval from 3k to 3k+1 (for k = 0, 1, 2) the curve is concave down, rising from height 2 to height 5; on each interval from 3k+1 to 3k+3 the curve is concave up, descending from height 5 back to height 2. The six labelled points (0, 2), (1, 5), (3, 2), (4, 5), (6, 2), (7, 5) are visible on the graph.
Graph of f

The graph of a periodic function space f is shown above for 0 less or equal than x less or equal than 7. The function space f is defined for all real numbers and has period 3.

Find the value of space f open parentheses 100 close parentheses. Show the work that leads to your answer.

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12b
1 mark

Find the average rate of change of space f on the interval 13 less or equal than x less or equal than 15. Show the work that leads to your answer.

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12c
1 mark

On the interval open parentheses 43 comma 44 close parentheses, describe the concavity of the graph of space f and determine whether the rate of change of space f is increasing or decreasing.

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