AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsInverse & Reciprocal Trigonometric Functions

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

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Inverse & Reciprocal Trigonometric Functions

1
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1 mark

The function space f models the horizontal displacement, in centimeters, of a point on the edge of a rotating wheel from its center vertical axis as a function of the wheel's angle of rotation theta, in radians. Which of the following best describes the meaning of space f to the power of negative 1 end exponent open parentheses 2.5 close parentheses in this context?

  • The angle of rotation, in radians, for which the point has a horizontal displacement of 2 . 5 centimeters

  • The horizontal displacement, in centimeters, of the point when the angle of rotation is 2 . 5 radians

  • The reciprocal of the horizontal displacement, in centimeters, of the point when the angle of rotation is 2 . 5 radians

  • The time, in seconds, it takes for the wheel to rotate such that the point has a horizontal displacement of 2 . 5 centimeters

Choose your answer

2
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1 mark

What is the exact value of cot open parentheses fraction numerator 7 pi over denominator 6 end fraction close parentheses?

  • - \sqrt{3}

  • - \frac{\sqrt{3}}{3}

  • \frac{\sqrt{3}}{3}

  • \sqrt{3}

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3
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1 mark

What is the exact value of csc open parentheses pi over 6 close parentheses?

  • \dfrac{1}{2}

  • \dfrac{\sqrt{3}}{3}

  • \dfrac{\sqrt{3}}{2}

  • 2

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4
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1 mark

What is the exact value of cos to the power of negative 1 end exponent open parentheses negative 1 half close parentheses?

  • - \frac{\pi}{3}

  • \frac{\pi}{3}

  • \frac{2 \pi}{3}

  • \frac{4 \pi}{3}

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5
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1 mark

What is the exact value of cos to the power of negative 1 end exponent open parentheses sin open parentheses fraction numerator 4 pi over denominator 3 end fraction close parentheses close parentheses ?

  • - \frac{\pi}{6}

  • \frac{5 \pi}{6}

  • \frac{7 \pi}{6}

  • \frac{4 \pi}{3}

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6
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1 mark

An angle \theta is in standard position in the xy-plane. It is known that sin theta equals 5 over 13 and the terminal ray of theta lies in Quadrant II. What is the exact value of cot theta ?

  • -\frac{12}{5}

  • -\frac{5}{12}

  • \frac{5}{12}

  • \frac{12}{5}

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7
1 mark
Graph of the cotangent function with vertical asymptotes at x = -2π,-π, 0, π and 2π; y-axis labelled -4 to 4, x-axis labelled -2π to 2π.
Graph of f

The figure above shows the graph of the function space f given by space f left parenthesis x right parenthesis equals cot left parenthesis x right parenthesis. Which of the following statements about the graph of space f is true?

  • The graph has vertical asymptotes for domain values where tan left parenthesis x right parenthesis equals 0, and space f is strictly decreasing between consecutive asymptotes.

  • The graph has vertical asymptotes for domain values where cos left parenthesis x right parenthesis equals 0, and space f is strictly decreasing between consecutive asymptotes.

  • The graph has vertical asymptotes for domain values where tan left parenthesis x right parenthesis equals 0, and space f is strictly increasing between consecutive asymptotes.

  • The range of f is open parentheses negative infinity comma negative 1 close square brackets union open square brackets 1 comma infinity close parentheses, and space f is strictly increasing between consecutive asymptotes.

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8
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1 mark

What is the exact value of cos open parentheses sin to the power of negative 1 end exponent open parentheses 3 over 5 close parentheses close parentheses?

  • \frac{3}{5}

  • \frac{3}{4}

  • \frac{4}{5}

  • \frac{5}{4}

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9
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1 mark

What is the exact value of tan open parentheses arcsin open parentheses 3 over 5 close parentheses close parentheses?

  • \dfrac{3}{5}

  • \dfrac{3}{4}

  • \dfrac{4}{5}

  • \dfrac{4}{3}

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10
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1 mark

If sec theta equals negative 5 over 4 and pi over 2 less than theta less than pi, what is the value of \cot \theta?

  • - \dfrac{4}{3}

  • - \dfrac{3}{4}

  • \dfrac{3}{4}

  • \dfrac{4}{3}

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11
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1 mark

What is the exact value of sec open parentheses arctan open parentheses square root of 3 close parentheses close parentheses?

  • 1 half

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

  • square root of 3

  • 2

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12
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1 mark
Graph of a sine wave with peaks at (-4, 3), (0, 3) and (4, 3), and troughs at (-2, -1), (2, -1) and (6, -1), oscillating along the x-axis from x=-5 to x=7, showing cycles and symmetry.
Graph of f

The figure above shows the graph of the sinusoidal function space f left parenthesis x right parenthesis equals a cos left parenthesis b x right parenthesis plus d, where a, b, and d are constants. The function g is given by g left parenthesis x right parenthesis equals a sec left parenthesis b x right parenthesis plus d. Which of the following gives the locations of the vertical asymptotes of the graph of g in the xy-plane?

  • x = 2k, for any integer k

  • x = 1 + 2k, for any integer k

  • x = \frac{4}{3} + 4k and x = \frac{8}{3} + 4k, for any integer k

  • x = 2 + 4k, for any integer k

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13
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 2 cos open parentheses 3 x close parentheses and has a period of fraction numerator 2 pi over denominator 3 end fraction. In order to define the inverse function of space f, which of the following specifies a restricted domain for space f and provides a rationale for why space f is invertible on that domain?

  • 0 less or equal than x less or equal than pi over 3, because all possible values of space f open parentheses x close parentheses occur without repeating on this interval

  • negative pi over 6 less or equal than x less or equal than pi over 6, because all possible values of space f open parentheses x close parentheses occur without repeating on this interval

  • 0 less or equal than x less or equal than pi over 3, because the length of this interval is half of the period

  • negative pi over 6 less or equal than x less or equal than pi over 6, because the length of this interval is half of the period

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14
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1 mark

An angle theta in standard position satisfies 0 less or equal than theta less than 2 pi. The terminal ray of theta passes through the point open parentheses negative 7 comma 4 close parentheses. What is the value of \theta, expressed in radians as a decimal approximation?

  • 0 . 519

  • 2 . 622

  • 3 . 661

  • 5 . 764

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15
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1 mark

Which of the following is equivalent to sin open parentheses 2 arctan x close parentheses?

  • \dfrac{x}{x^{2} + 1}

  • \dfrac{2 x}{x^{2} + 1}

  • \dfrac{2 x}{1 - x^{2}}

  • \dfrac{x^{2} - 1}{x^{2} + 1}

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16
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1 mark

If csc theta equals negative 5 over 3 and \theta is in quadrant III, what is the value of \cot \theta?

  • \dfrac{3}{4}

  • \dfrac{4}{3}

  • \dfrac{5}{4}

  • \dfrac{5}{3}

Choose your answer

17
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1 mark

What is the value of sec open parentheses arctan open parentheses 2 close parentheses close parentheses?

  • \dfrac{1}{\sqrt{5}}

  • \dfrac{\sqrt{5}}{2}

  • \sqrt{3}

  • \sqrt{5}

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