# How do you evaluate the inverse function by sketching a unit circle, locating the correct angle, and evaluating the ordered pair on the circle for: #tan^-1 (0)# and #csc^-1 (2)#?

##### 1 Answer

The trigonometric functions (

The inverse trigonometric functions (

Let us take a look at a unit circle diagram:

We will start with

#y/x = 0# .

Clearly, this statement can only be true if

So,

#arctan 0 = 0# .

Let us move on to

Well, the cosecant of an angle is the inverse of its sine. In other words,

#csc theta = 1/sin theta# .

We know that sine gives a ratio between the opposite side and the hypotenuse. So, the cosecant function therefore gives a ratio between the hypotenuse and the opposite side. And, if the arc-cosecant takes this ratio as an argument, and gives the angle, then we know that

#2 = r/y#

This is more conveniently written as:

#2y = r#

Or, alternatively as:

#y = 1/2 r#

What this tells us is that for our angle

And, elementary geometry tells us that this is precisely what occurs in a 30-60-90 triangle.

If