STUFFFFFFFFSS ;)

One of the more common notations for inverse trig functions can be very confusing. First, regardless of how you are used to dealing with exponentiation we tend to denote an inverse trig function with an “exponent” of “-1”. In other words, the inverse cosine is denoted as . It is important here to note that in this case the “-1” is NOT an exponent and so,

.

In inverse trig functions the “-1” looks like an exponent but it isn’t, it is simply a notation that we use to denote the fact that we’re dealing with an inverse trig function. It is a notation that we use in this case to denote inverse trig functions. If I had really wanted exponentiation to denote 1 over cosine I would use the following.

There’s another notation for inverse trig functions that avoids this ambiguity. It is the following.

So, be careful with the notation for inverse trig functions!

There are, of course, similar inverse functions for the remaining three trig functions, but these are the main three that you’ll see in a calculus class so I’m going to concentrate on them.

To evaluate inverse trig functions remember that the following statements are equivalent.

In other words, when we evaluate an inverse trig function we are asking what angle, , did we plug into the trig function (regular, not inverse!) to get x.

So, let’s do some problems to see how these work. Evaluate each of the following.

1.

In other words, we asked what angles, t, do we need to plug into cosine to get ? This is essentially what we are asking here when we are asked to compute the inverse trig function.

There is one very large difference however. In Problem 1 we were solving an equation which yielded an infinite number of solutions. These were,

In the case of inverse trig functions we are after a single value. We don’t want to have to guess at which one of the infinite possible answers we want. So, to make sure we get a single value out of the inverse trig cosine function we use the following restrictions on inverse cosine.

The restriction on the guarantees that we will only get a single value angle and since we can’t get values of x out of cosine that are larger than 1 or smaller than -1 we also can’t plug these values into an inverse trig function.

So, using these restrictions on the solution to Problem 1 we can see that the answer in this case is