AP PreCalculus Practice Quiz: The Secant, Cosecant, and Cotangent Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 11 questions to check your progress.
Question 1 of 11
All Questions (11)
A) Cosecant function
B) Secant function
C) Cotangent function
D) Tangent function
Correct Answer: B
According to the provided content, 'The secant function, f(θ) = sec θ, is the reciprocal of the cosine function'.
A) Sine function
B) Cosine function
C) Tangent function
D) Secant function
Correct Answer: A
The provided content states that 'The cosecant function, f(θ) = csc θ, is the reciprocal of the sine function'.
A) When sin θ = 0
B) When cos θ = 1
C) When cos θ = 0
D) When sin θ = 1
Correct Answer: C
The definition of the secant function is given as the reciprocal of the cosine function, with the specific condition that it is defined 'where cos θ ≠ 0'. Therefore, the function is undefined when cos θ = 0.
A) sin θ = 1
B) cos θ = 0
C) sin θ = 0
D) cos θ = 1
Correct Answer: C
The content explicitly states that 'The graphs of the secant and cosecant functions have vertical asymptotes where cosine and sine are zero, respectively'. Thus, the cosecant function has vertical asymptotes where sin θ = 0.
A) 0.5
B) 0
C) -0.8
D) -2
Correct Answer: D
The provided content specifies that the range of the secant and cosecant functions is (–∞, -1] ∪ [1, ∞). Of the options given, only -2 falls within this range.
A) (sin θ) / (cos θ)
B) (cos θ) / (sin θ)
C) 1 / (cos θ)
D) 1 / (sin θ)
Correct Answer: B
The cotangent function is the reciprocal of the tangent function. Since the tangent function is the quotient (sin θ) / (cos θ), its reciprocal, the cotangent function, must be the quotient (cos θ) / (sin θ).
A) 5
B) -5
C) 1/5
D) The value cannot be determined.
Correct Answer: B
The secant function is the reciprocal of the cosine function. Therefore, sec θ = 1 / (cos θ). If cos θ = -1/5, then sec θ = 1 / (-1/5) = -5.
A) When cos θ = 0
B) When tan θ = 1
C) When sin θ = 0
D) When tan θ = 0
Correct Answer: D
The content states that 'The cotangent function, f(θ) = cot θ, is the reciprocal of the tangent function, where tan θ ≠ 0'. This means the cotangent function is undefined when tan θ = 0.
A) They are both reciprocals of the sine function.
B) They both have vertical asymptotes where cos θ = 0.
C) They both have a range of (–∞, -1] ∪ [1, ∞).
D) They are both defined for all real numbers.
Correct Answer: C
The provided content explicitly states that the graphs of both the secant and cosecant functions 'have a range of (–∞, -1] ∪ [1, ∞)'. While they both have vertical asymptotes, they occur at different locations (where cosine is zero for secant, and where sine is zero for cosecant).
A) -2/3
B) -3/2
C) 3/2
D) The value is outside the function's range.
Correct Answer: C
The cosecant function is the reciprocal of the sine function. Therefore, csc θ = 1 / (sin θ). If sin θ = 2/3, then csc θ = 1 / (2/3) = 3/2. This value is in the range [1, ∞).
A) Where sin θ = 0, because sec θ = csc(π/2 - θ)
B) Where cos θ = 0, because sec θ = 1 / cos θ
C) Where tan θ = 0, because secant is related to tangent
D) Where sin θ = 1, because that is a maximum value
Correct Answer: B
The secant function is defined as the reciprocal of the cosine function, meaning sec θ = 1 / cos θ. As a quotient, its denominator is cos θ. Therefore, vertical asymptotes will occur where the denominator is zero, which is where cos θ = 0. This is consistent with the provided content.