Question

Which of the following trigonometric functions contain undefined value(s) from [0, 2π)? Select all that apply.

A) sin θ
B) cos θ
C) tan θ
D) csc θ
E) sec θ
F) cot θ

Answer 1

The trigonometric functions with undefined values from [0, 2π) are tan θ, csc θ, sec θ, and cot θ, as they involve division by zero at certain angles.

Step-by-step explanation:

The trigonometric functions that contain undefined values from [0, 2π) are tan θ, csc θ, sec θ, and cot θ. These functions are undefined for certain angles in the range of [0, 2π) because they involve division by zero. For instance, tan θ = sin θ / cos θ is undefined whenever cos θ = 0, which occurs at θ = π/2 and 3π/2. Similarly, csc θ = 1/sin θ is undefined when sin θ = 0, i.e., at θ = 0, π, and 2π. The sec θ = 1/cos θ and cot θ = cos θ/sin θ also become undefined when their denominators are zero. The sin θ and cos θ functions do not have points of undefined values in the given range because they are ratios of the sides of a right-angled triangle and will always have values between -1 and 1.