Question

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

Short answer

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

Step-by-step solution

Step 1. Given Information 

We have to find for what numbers do$cscx=1andcscx=-1.$

Step 2. Finding the numbers 

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

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

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

Thus, $cscx=1$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 role="math" localid="1646400679548" $-\frac{\mathrm{\pi }}{2}and\frac{3\mathrm{\pi }}{2}$in the given interval.

Thus, the cosecant function is $-1$at role="math" localid="1646400699176" $-\frac{\mathrm{\pi }}{2}and\frac{3\mathrm{\pi }}{2}$in the given interval.