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 (##"sin"##, ##"cos"##, ##"tan"##) all take angles as their arguments, and produce ratios. (remember SOHCAHTOA)

The inverse trigonometric functions (##"arcsin"##, ##"arccos"##) take ratios as their arguments, and produce the corresponding angles.

Let us take a look at a unit circle diagram:

##r## is the radius of the circle, and it is also the hypotenuse of the right triangle.

We will start with ##arctan 0##. First, we know that the tangent of an angle equals the ratio between the opposite side and the adjacent side. And, we know that the arc tangent function takes a ratio of this form, and produces an angle. Since ##0## is our arc tangent's argument, then it must be equal to the ratio:

##y/x = 0##.

Clearly, this statement can only be true if ##y = 0##. And if ##y= 0##, then ##theta## must also be ##0##.

So,

##arctan 0 = 0##.

Let us move on to ##"arccsc"(2)##.

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## must be the ratio between the hypotenuse and the opposite side.

##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 ##theta## to equal the ##"arccsc"## of ##2##, we need a right triangle whose hypotenuse is twice the length of its opposite leg.

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

If ##r = 2y##, then ##x = ysqrt(3)##. Therefore, ##theta## is equal to ##30## degrees, or ##pi/6##.