Question

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)`?

Answer 1

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`.