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
Aug 4, 2014

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:
Unit circle diagram with radius

#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#.