How do you know if csc xcscx is an even or odd function?

1 Answer
Dec 21, 2015

Odd.

Explanation:

Know that sin(-x)=-sin(x)sin(x)=sin(x).

Even function: when f(-x)=f(x)f(x)=f(x)
Odd function: when f(-x)=-f(x)f(x)=f(x)

In this case, f(x)=cscxf(x)=cscx

f(x)=1/sinxf(x)=1sinx

f(-x)=1/(sin(-x))f(x)=1sin(x)

f(-x)=1/(-sinx)f(x)=1sinx

f(-x)=-cscx=-f(x)f(x)=cscx=f(x)

Thus, the function is odd.
graph{csc(x) [-10, 10, -5, 5]}
A property of odd functions is that they have origin symmetry, which means the graph can is symmetrical when reflected over the point (0,0)(0,0).