Question

What is the domain of validity for csc theta = 1 / sin theta?
a. All real numbers
b. All real numbers except odd multiples of pi/ 2
c. All real numbers except even multiples of pi/2
d. All real numbers except multiples of pi.

Answer 1

$De finition\ of\ csc\theta=\frac{1}{sin\theta}\\-----------------------\\therefore\ csc\theta=\frac{1}{sin\theta}\to\frac{1}{sin\theta}=\frac{1}{sin\theta}\\\\the\ denominator\ must\ be\ different\ from\ zero,\ therefore\\\\sin\theta\neq0\iff\theta\neq k\pi\ where\ k\in\mathbb{Z}\\\\Answer:\boxed{D}$