Inverse Functions
Inverse trig functions are sort of like bizarro trig functions. They probably were meant to live in a parallel universe, but somehow they ended up here. Now we're stuck with them. We guess we should learn how to handle them since they're here to stay.
Let's go over their notation first. Inverse trig functions are indicated with an exponent of -1.
- Inverse sine is written sin-1 x.
- Inverse cosine is written cos-1 x.
- Inverse tangent is written tan-1 x.
Similarly,
- Inverse cosecant is written csc-1 x.
- Inverse secant is written sec-1 x.
- Inverse cotangent is written cot-1 x.
This "-1" is not a true exponent. It's just a convenient way to indicate that we're using the inverse functions. We told you they were bizarre.
Just to be clear, 
You may have seen another notation for inverse functions using the prefix "arc."
arcsin x = sin-1 x
arccos x = cos-1 x
arctan x = tan-1 x
arccsc x = csc-1 x
arcsec x = sec-1 x
arccot x = cot-1 x
When working with inverse trig functions, here's how it works:
sin ɵ = x and ɵ = sin-1 x
cos ɵ = x and ɵ = cos-1 x
tan ɵ = x and ɵ = tan-1 x
csc ɵ = x and ɵ = csc-1 x
sec ɵ = x and ɵ = sec-1 x
cot ɵ = x and ɵ = cot-1 x
The normal trig functions give us a number when we plug in an angle, but the inverse trig functions spit out an angle when we plug in a number.
For example, say we're trying to find the value of sin-1(1). Since that's the inverse sine, we're looking for the angle whose sine is 1. Thinking back to our unit circle, we know the sine has a value of positive 1 when the angle is 90°, or π⁄2 radians. In other words, since we know that sin(90°) = 1, that must mean:
sin-1(1) = 90°
One more thing: inverse trig functions are "restricted" or "bounded" to specific domains (or x-coordinates).
We'll be able to see those restrictions for sine and cosine by checking out the graphs of their inverse functions.