AP PreCalculus Practice Quiz: Inverse Trigonometric Functions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

Why must the domains of the sine, cosine, and tangent functions be restricted to define their inverse functions?

A)Because the trigonometric functions are periodic, they fail the horizontal line test over their entire domain.B)Because the trigonometric functions have vertical asymptotes, which limits their range.C)Because the range of the trigonometric functions is limited to [-1, 1].D)Because the inverse trigonometric functions must be defined for all real numbers.

All Questions (10)

Why must the domains of the sine, cosine, and tangent functions be restricted to define their inverse functions?

A) Because the trigonometric functions are periodic, they fail the horizontal line test over their entire domain.

B) Because the trigonometric functions have vertical asymptotes, which limits their range.

C) Because the range of the trigonometric functions is limited to [-1, 1].

D) Because the inverse trigonometric functions must be defined for all real numbers.

Correct Answer: A

The content states that 'Because the corresponding trigonometric functions are periodic, they are only invertible if they have restricted domains.' A periodic function repeats its y-values at regular intervals, meaning it is not one-to-one and would fail the horizontal line test, which is the graphical test for invertibility.

To define the inverse sine function, arcsin(x), the domain of the sine function is restricted to which of the following intervals?

A) [0, π]

B) (-π/2, π/2)

C) [-π/2, π/2]

D) (-∞, ∞)

Correct Answer: C

The provided content explicitly states, 'The domain of the sine function is restricted to [−π/2, π/2]... to define [its] respective inverse functions.'

What is the restricted domain of the cosine function used to define the arccosine function?

A) [-π/2, π/2]

B) [0, π]

C) (0, π)

D) (-π/2, π/2)

Correct Answer: B

According to the provided text, 'the cosine function [is restricted] to [0, π]... to define [its] respective inverse functions.'

The arctangent function is defined by restricting the domain of the tangent function to which interval?

A) [-π/2, π/2]

B) [0, π]

C) (0, π)

D) (-π/2, π/2)

Correct Answer: D

The content specifies that the domain of the tangent function is restricted to (-π/2, π/2). The use of parentheses indicates an open interval, which is a key distinction for the tangent function due to its vertical asymptotes at x = -π/2 and x = π/2.

Which of the following is an alternative name for the inverse cosine function?

A) Secant

B) Arcsine

C) Arccosine

D) Cosecant

Correct Answer: C

The content states, 'The inverse trigonometric functions are called arcsine, arccosine, and arctangent.' Arccosine is the name for the inverse cosine function.

In the expression y = arcsin(x), what do the variables x and y represent?

A) x is an angle measure, and y is the sine of that angle.

B) x is a value in the range of sine, and y is an angle measure.

C) Both x and y are angle measures.

D) Both x and y are ratios from a right triangle.

Correct Answer: B

The content explains that 'The output value of an inverse trigonometric function is often interpreted as an angle measure and the input is a value in the range of the corresponding trigonometric function.' Therefore, for y = arcsin(x), x is the input value (from the range of sine, i.e., [-1, 1]) and y is the output angle.

The output of the function y = arccos(x) must be an angle in which interval?

A) [0, π]

B) (-π/2, π/2)

C) [-π/2, π/2]

D) [0, 2π]

Correct Answer: A

The range of an inverse function is the restricted domain of the original function. Since the domain of the cosine function is restricted to [0, π] to make it invertible, the range (the set of possible output values) of the arccosine function is [0, π].

A student calculates a value for arcsin(-0.5) and gets 11π/6. Why is this answer incorrect based on the standard definition of the arcsine function?

A) The input to arcsine must be positive.

B) The output of arcsine must be in the interval [0, π].

C) The output of arcsine must be in the interval [-π/2, π/2].

D) The calculation is correct; 11π/6 is coterminal with the correct angle.

Correct Answer: C

The domain of the sine function is restricted to [-π/2, π/2] to define its inverse. Consequently, the range (output) of the arcsine function is [-π/2, π/2]. The value 11π/6 is outside this interval. The correct value is -π/6, which is in the required interval and has a sine of -0.5.

Which of the following values could NOT be an output of the y = arctan(x) function?

A) π/2

B) π/4

C) 0

D) -π/3

Correct Answer: A

The domain of the tangent function is restricted to the open interval (-π/2, π/2) to define the arctangent function. This means the range of arctan(x) is also the open interval (-π/2, π/2). The value π/2 is an endpoint and is not included in the open interval, so it cannot be an output of the function.

The graph of y = arccos(x) is constructed by reflecting a portion of the graph of y = cos(x) across the line y = x. Which portion of the y = cos(x) graph is used?

A) The portion over the interval x ∈ [-π/2, π/2].

B) The entire graph of y = cos(x).

C) The portion over the interval x ∈ (-π/2, π/2).

D) The portion over the interval x ∈ [0, π].

Correct Answer: D

The content states that an inverse function is constructed over a restricted domain. For the cosine function, this restricted domain is [0, π]. The graph of an inverse function is the reflection of the original function's graph (on its restricted domain) across the line y = x. Therefore, only the part of y = cos(x) from x = 0 to x = π is used.