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AP Precalculus -3.11 Secant, Cosecant, and Cotangent- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -3.11 Secant, Cosecant, and Cotangent- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -3.11 Secant, Cosecant, and Cotangent- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The function \( f \) is defined by \( f(x) = \sec\left(\frac{1}{2}\left(x – \frac{\pi}{2}\right)\right) \). Which of the following describes the domain of \( f \)?
(A) The domain is the set of all real numbers \( x \), except when \( x = \frac{\pi}{2} + n\pi k \), where \( k \) is any integer.
(B) The domain is the set of all real numbers \( x \), except when \( x = \frac{\pi}{2} + 2n\pi k \), where \( k \) is any integer.
(C) The domain is the set of all real numbers \( x \), except when \( x = \pi + 2n\pi k \), where \( k \) is any integer.
(D) The domain is the set of all real numbers \( x \), except when \( x = \frac{3\pi}{2} + 2n\pi k \), where \( k \) is any integer.
▶️ Answer/Explanation
Detailed solution

\( f(x) = \sec\left(\frac12\left(x – \frac{\pi}{2}\right)\right) \) is undefined where \( \cos\left(\frac12\left(x – \frac{\pi}{2}\right)\right) = 0 \).
Let \( u = \frac12\left(x – \frac{\pi}{2}\right) \). Cosine is zero when \( u = \frac{\pi}{2} + n\pi \).
Thus \( \frac12\left(x – \frac{\pi}{2}\right) = \frac{\pi}{2} + n\pi \)
Multiply by 2: \( x – \frac{\pi}{2} = \pi + 2n\pi \)
So \( x = \frac{\pi}{2} + \pi + 2n\pi = \frac{3\pi}{2} + 2n\pi \).
That’s \( x = \frac{3\pi}{2} + 2\pi k \) (letting \( k = n \)).
✅ Answer: (D)

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