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

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2
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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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3
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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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4
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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5
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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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6
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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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