On the unit circle, which trigonometric functions are undefined when $x=0$?

• A. $\sin$ and $\tan$
• B. $\tan$ and $\sec$
• C. $\sin$ and $\csc$
• D. $\cos$ and $\cot$

Answer: (B)

Verified step by step guidance

1. Recall that on the unit circle, the angle \( x = 0 \) corresponds to the point \( (\cos 0, \sin 0) = (1, 0) \).
2. Evaluate the basic trigonometric functions at \( x = 0 \): \( \sin 0 = 0 \) and \( \cos 0 = 1 \).
3. Understand that the tangent function is defined as \( \tan x = \frac{\sin x}{\cos x} \). Since \( \cos 0 = 1 \), \( \tan 0 = 0 \) is defined.
4. Recall that the secant function is defined as \( \sec x = \frac{1}{\cos x} \). Since \( \cos 0 = 1 \), \( \sec 0 = 1 \) is defined.
5. Check the cosecant and cotangent functions: \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Since \( \sin 0 = 0 \), both \( \csc 0 \) and \( \cot 0 \) are undefined.