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)#?
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
#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.