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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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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2
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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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3
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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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4
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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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5
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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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6
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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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