Question

Determine whether each equation is true for all x for which both sides of the equation are defined. If it is true, support your conclusion with a sketch using the unit circle. If it is false, give a counterexample.
inverse sin(-x)=-inverse sin(x)
inverse cos(-x)=-inverse cos(x)

I think that the first one is true and the second one is false but I do not know what to put down as a sketch or counterexample. Any help is appreciated.

Answer 1

The simplest way to determine if
inverse f(-x) =-inverse f(x)
is true for all values of x is to find out if f(x)=f(-x) for all values of x.

Substitute sin(x) or cos(x) for f(x) above and do a little sketching of each of the two trigonometric functions between -pi and +pi. You will get your answer by simply looking at the sketches.