Question

which function is undefined when $\theta=\frac{\pi}{2}$ radians?
\\(\cos \theta\\)
\\(\cot \theta\\)
\\(\csc \theta\\)
\\(\tan \theta\\)

Explanation:

Step1: Recall Trig Function Definitions

• $\cos\theta$: Defined for all real $\theta$, $\cos\frac{\pi}{2} = 0$.
• $\cot\theta=\frac{\cos\theta}{\sin\theta}$: $\sin\frac{\pi}{2}=1$, so $\cot\frac{\pi}{2}=\frac{0}{1}=0$ (defined).
• $\csc\theta=\frac{1}{\sin\theta}$: $\sin\frac{\pi}{2}=1$, so $\csc\frac{\pi}{2}=\frac{1}{1}=1$ (defined).
• $\tan\theta=\frac{\sin\theta}{\cos\theta}$: $\cos\frac{\pi}{2}=0$, division by zero is undefined, so $\tan\frac{\pi}{2}$ is undefined.

Answer:

$\tan\theta$ (the option with "tan θ")