Question

If csc (x-5pi)=3 then csc x=??

How to get the answer and what is the concept behind this?

Answer 1

first recall that csc = 1/sin
\[\sin(x-5\pi) = \frac{1}{3}\]
next use angle sum identity
\[\sin(a-b) = \sin a \cos b - \sin b \cos a\]
\[\rightarrow \sin x \cos 5\pi - \sin 5\pi \cos x = \frac{1}{3}\]
from unit circle we know that
cos(5pi) = cos(pi) = -1
sin(5pi) = sin(pi) = 0
\[\rightarrow \sin x (-1) - 0 = \frac{1}{3}\]
\[\sin x = -\frac{1}{3}\]
\[\csc x = -3\]