Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.9 Trigonometric Functions
Forms
Recall that trigonometric functions take an angle as an input and output a ratio between the sides of a triangle. An Inverse Trigonometric function will do the opposite, so it will take a ratio of sides, or a number, as an input and output the angle that would correspond to that value. In terms of notation, there are two main ways to write an inverse trigonometric function. The standard way uses the f-1 notation for an inverse. So, the inverse of sinθ would be sin-1x. However, this is terribly confusing because we can also describe cscx as sin-1x because 
, if we think of the -1 as an exponent.
In light of this terrible notation, we will opt instead for the second way to denote an inverse trig function: by placing ‘arc’ in front the function whose inverse we are evaluation. For example, arcsinx will be the inverse of sinx. Not only is this much more clear, but it also sounds more impressive too!
Here is a graph of all six inverse trigonometric functions. The colors correspond to the colors of the trigonometric functions in the previous section. For example, blue is arccos(x).
For the sake of clarity, here is each of them separated:
arcsinx
arctanx
arcsecx
arccosx
arccotx
arccscx
To evaluate an inverse trigonometric function, it is often helpful to convert it to its trigonometric counterpart. It is generally accepted there is one main answer, but it is helpful to recall that on the unit circle there are many values that repeat, so we will identify each possible answer when we can:
Example 1: For what angles between 0 and 2π does cosθ=1/2?
Solution: Using the unit circle, we find the x-coordinate(s) whose value equals 1/2 and the angle at which that occurs is the answer:
θ=π/3,5π/3
Example 2: Evaluate arccos(1/2).
Solution: This question is very similar to the previous, except when using inverse trigonometry we have to be very careful of the range. As is visible on the graph of arccos(x), we can see the domain is x∈[-1,1] and the range is y∈[0,π].
That means that of the two results from the previous example, θ=π/3,5π/3, only θ=π/3 is an acceptable answer because 5π/3∉[0,π].
Example 3: Evaluate arctan(√3)
Solution: For arctan(√3), we say that tanθ=√3. So,
tanθ=√3/1
This can be constructed from the angles θ=π/3 and 4π/3