AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsTrigonometric Equations, Inequalities & Identities

Trigonometric Equations, Inequalities & Identities (College Board AP® Precalculus): Exam Questions

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Trigonometric Equations, Inequalities & Identities

1
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1 mark

What is the smallest positive value of x for which csc left parenthesis x right parenthesis equals 0.5 x squared ?

  • 1 . 404

  • 1 . 422

  • 1 . 571

  • 2 . 902

Choose your answer

2
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1 mark

What is the value of x in the interval open square brackets 0 comma pi over 2 close square brackets for which 2 sin x equals square root of 2?

  • pi over 6

  • pi over 4

  • pi over 3

  • pi over 2

Choose your answer

3
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1 mark

Which of the following is equivalent to cos open parentheses alpha minus beta close parentheses?

  • cos alpha cos beta plus sin alpha sin beta

  • cos alpha cos beta minus sin alpha sin beta

  • sin alpha cos beta minus cos alpha sin beta

  • sin alpha cos beta plus cos alpha sin beta

Choose your answer

4
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1 mark

The function h is given by h \left(x\right) = 5 \cos \left(2 x\right). Which of the following is an equivalent form for h \left(x\right) ?

  • h \left(x\right) = 10 \cos x \sin x

  • h \left(x\right) = 5 - 10 \sin^{2} x

  • h \left(x\right) = 5 \cos^{2} x + 5 \sin^{2} x

  • h \left(x\right) = \left(5 \cos x\right) \left(5 \sin x\right)

Choose your answer

5
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1 mark

What are all values of \theta, for 0 \leq \theta < 2 \pi, where 2 \cos^{2} \theta = \cos \theta?

  • \frac{\pi}{3} , \frac{\pi}{2} , \frac{3 \pi}{2} , \text{ and } \frac{5 \pi}{3}

  • \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{2 \pi}{3} , \text{ and } \frac{4 \pi}{3}

  • 0 , \pi , \frac{\pi}{3} , \text{ and } \frac{5 \pi}{3}

  • 0 , \pi , \frac{\pi}{6} , \text{ and } \frac{11 \pi}{6}

Choose your answer

6
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1 mark

The function f is given by f \left(x\right) = \cos^{2} \left(0.5 x + 0.3\right). The function g is given by g \left(x\right) = f \left(x\right) - 0.5. What are the zeros of g on the interval 0 \leq x \leq 2 \pi ?

  • 0.971 and 4.112

  • 1.494

  • 0.447 and 4.636

  • 2.542

Choose your answer

7
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1 mark

The function f is given by f open parentheses x close parentheses equals sec squared open parentheses 0.5 x plus 0.4 close parentheses. The function g is given by g open parentheses x close parentheses equals f open parentheses x close parentheses minus 2. What are the zeros of g on the interval 0 \leq x \leq 2 \pi ?

  • 0 . 247 and 4 . 436

  • 0 . 771 and 3 . 912

  • 1 . 294

  • 2 . 342

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8
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1 mark

Let f(x) = 2 + 4\csc x and g(x) = -6. In the xy-plane, what are the x-coordinates of the points of intersection of the graphs of f and g for 0 < x < 2\pi?

  • x = \frac{\pi}{6} \textrm{ and } x = \frac{5\pi}{6}

  • x = \frac{4\pi}{3} \textrm{ and } x = \frac{5\pi}{3}

  • x = \frac{5\pi}{6} \textrm{ and } x = \frac{7\pi}{6}

  • x = \frac{7\pi}{6} \textrm{ and } x = \frac{11\pi}{6}

Choose your answer

9
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1 mark

For -\pi < \theta < \pi, which of the following gives all values of \theta for which 2 cos theta greater than negative 1 and 2 sin theta greater than negative square root of 3?

  • -\frac{\pi}{3} < \theta < \frac{\pi}{3}

  • -\frac{2\pi}{3} < \theta < \frac{2\pi}{3}

  • -\frac{\pi}{3} < \theta < \frac{2\pi}{3}

  • -\frac{2\pi}{3} < \theta < \frac{\pi}{3}

Choose your answer

10
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1 mark

The function h is given by h\left(x\right) = 3\cos\left(2x\right). Which of the following is an equivalent form for h\left(x\right)?

  • h\left(x\right) = 6\cos x\sin x

  • h\left(x\right) = 6\cos^{2} x - 3

  • h\left(x\right) = 3\cos^{2} x + 3\sin^{2} x

  • h\left(x\right) = \left(3\cos x\right)\left(3\sin x\right)

Choose your answer

11
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1 mark

What are all values of \theta, for 0 \leq \theta < 2 \pi, where 2 \sin^{2} \theta = \sin \theta?

  • 0 , \frac{\pi}{6} , \frac{5 \pi}{6} , \textrm{ }\text{and}\textrm{ } \pi

  • 0 , \pi , \frac{7 \pi}{6} , \textrm{ }\text{and}\textrm{ } \frac{11 \pi}{6}

  • \frac{\pi}{6} , \frac{\pi}{2} , \frac{5 \pi}{6} , \textrm{ }\text{and}\textrm{ } \frac{3 \pi}{2}

  • \frac{\pi}{3} , \frac{\pi}{2} , \frac{2 \pi}{3} , \textrm{ }\text{and}\textrm{ } \frac{3 \pi}{2}

Choose your answer

12
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1 mark

An angle \theta is in standard position. If sin theta equals negative 2 over 3 and pi less than theta less than fraction numerator 3 pi over denominator 2 end fraction, what is the exact value of sec theta ?

  • \frac{3 \sqrt{5}}{5}

  • - \frac{\sqrt{5}}{3}

  • - \frac{3 \sqrt{5}}{5}

  • - \frac{3}{2}

Choose your answer

13
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1 mark

What is the exact value of cos open parentheses fraction numerator 5 pi over denominator 12 end fraction close parentheses ?

  • fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction

  • fraction numerator square root of 6 plus square root of 2 over denominator 4 end fraction

  • fraction numerator square root of 2 minus square root of 6 over denominator 4 end fraction

  • fraction numerator square root of 6 over denominator 4 end fraction

Choose your answer

14
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1 mark

The expression fraction numerator 1 plus cos theta over denominator sin theta end fraction plus fraction numerator sin theta over denominator 1 plus cos theta end fraction is equivalent to which of the following?

  • 2 csc theta

  • 2 sec theta

  • csc theta

  • 2 cot theta

Choose your answer

15
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1 mark

What are all values of x, for 0 less or equal than x less than 2 pi, for which 2 sin x greater or equal than 1 ?

  • \frac{\pi}{6} \leq x \leq \frac{5 \pi}{6}

  • 0 \leq x < \frac{\pi}{6} or \frac{5 \pi}{6} < x < 2 \pi

  • \frac{\pi}{6} \leq x \leq \frac{11 \pi}{6}

  • \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3}

Choose your answer

16
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1 mark

Which of the following is a value of x in the interval open square brackets 0 comma pi close square brackets for which sin open parentheses 2 x close parentheses equals 1 half?

  • \dfrac{\pi}{6}

  • \dfrac{\pi}{4}

  • \dfrac{\pi}{3}

  • \dfrac{5 \pi}{12}

Choose your answer

17
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1 mark

For sin x not equal to 0 and cos x not equal to 1, a student attempts to simplify the expression fraction numerator sin x over denominator 1 minus cos x end fraction plus fraction numerator 1 minus cos x over denominator sin x end fraction using the following four steps:

Step 1. Combine over a common denominator:

fraction numerator sin x over denominator 1 minus cos x end fraction plus fraction numerator 1 minus cos x over denominator sin x end fraction equals fraction numerator sin squared x plus open parentheses 1 minus cos x close parentheses squared over denominator sin x open parentheses 1 minus cos x close parentheses end fraction

Step 2. Expand the squared term in the numerator:

equals fraction numerator sin squared x plus 1 minus cos squared x over denominator sin x open parentheses 1 minus cos x close parentheses end fraction

Step 3. Apply the Pythagorean identity \sin^{2} x = 1 - \cos^{2} x:

equals fraction numerator 2 open parentheses 1 minus cos squared x close parentheses over denominator sin x open parentheses 1 minus cos x close parentheses end fraction

Step 4. Factor 1 - \cos^{2} x = \left(1 - \cos x\right) \left(1 + \cos x\right) and cancel:

equals fraction numerator 2 open parentheses 1 plus cos x close parentheses over denominator sin x end fraction

In which step does the first mathematical error occur?

  • Step 1

  • Step 2

  • Step 3

  • Step 4

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18
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1 mark

Suppose sin A equals 3 over 5 where pi over 2 less than A less than pi, and cos B equals 5 over 13 where fraction numerator 3 pi over denominator 2 end fraction less than B less than 2 pi. What is the value of cos open parentheses A minus B close parentheses ?

  • - \frac{56}{65}

  • \frac{56}{65}

  • - \frac{16}{65}

  • \frac{16}{65}

Choose your answer

19
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1 mark

What are all values of x in the interval 0 less or equal than x less than 2 pi for which 2 sin squared x plus cos x equals 1 ?

  • x equals 0 comma pi over 3 comma fraction numerator 5 pi over denominator 3 end fraction

  • x equals 0 comma fraction numerator 2 pi over denominator 3 end fraction comma fraction numerator 4 pi over denominator 3 end fraction

  • x equals fraction numerator 2 pi over denominator 3 end fraction comma pi comma fraction numerator 4 pi over denominator 3 end fraction

  • x equals fraction numerator 2 pi over denominator 3 end fraction comma fraction numerator 4 pi over denominator 3 end fraction only

Choose your answer

20
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1 mark

If sin theta equals 3 over 5 and pi over 2 less than theta less than pi, what is the value of sin open parentheses 2 theta close parentheses?

  • - \dfrac{24}{25}

  • - \dfrac{7}{25}

  • \dfrac{7}{25}

  • \dfrac{24}{25}

Choose your answer

21
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1 mark

How many values of x in the interval open square brackets 0 comma 2 pi close parentheses satisfy 2 sin open parentheses 2 x minus pi over 3 close parentheses equals square root of 3?

  • 2

  • 4

  • 6

  • 8

Choose your answer

22
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1 mark

How many values of x in the interval open square brackets 0 comma 2 pi close parentheses satisfy sec x plus tan x equals 1?

  • 0

  • 1

  • 2

  • 3

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