Q. 12 - Precalculus Enhanced with Graphing Utilities

Question

For what numbers $x, -2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$ does $cscx=1?$ For what numbers x  does $cscx=-1?$

Step-by-Step Solution

Answer

For $x=\frac{\mathrm{\pi }}{2},-\frac{3\mathrm{\pi }}{2}, csc x =1$ and for $x=-\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}, csc x=-1.$

Step 1. Given Information

We have to find for what numbers do $csc x =1 and csc x =-1.$

Step 2. Finding the numbers

As we know the reciprocal of the cosecant function is $\frac{1}{\sin }.$

So, $csc x=1 when \sin x=1.$

The sine functions are $1$ at $\frac{\mathrm{\pi }}{2} and -\frac{3\mathrm{\pi }}{2}$ in the given interval.

Thus, $csc x =1$will be at $\frac{\mathrm{\pi }}{2} and -\frac{3\mathrm{\pi }}{2}.$

Step 3. Finding the numbers

Likewise, the cosecant function is $-1$, when sine function is $-1.$

The sine function is $-1$ at $-\frac{\mathrm{\pi }}{2} and \frac{3\mathrm{\pi }}{2}$ in the given interval.

Thus, the cosecant function is $-1$ at $-\frac{\mathrm{\pi }}{2} and \frac{3\mathrm{\pi }}{2}$ in the given interval.