We’re talking bout the inverse cosine function and I want to talk about some properties that inverse cosine has because it’s the inverse of the restricted cosine function. Remember the definition is y equals inverse cosine x means x equals cosine y as long as y is between zero and pi. Based on our definition these two guys are inverse functions. F of x equals cosine. This is the restricted cosine function and f inverse equals inverse cosine.
Remember for inverse functions you have two identities, this one and this one; f inverse of f of x equals x. So what would these be for these two functions. Well the first would be cosine of inverse cosine of x equals x and the second would be inverse cosine of x cosine x equals x. But you have to remember that these two identities are only true as long as x lives in the domain of the inside function. So in the case of this identity, x would have to be in the domain of inverse cosine which is from -1 to one. And for this one, x would have to be in the domain of the restricted cosine function. X would have to be between zero and pi.
Let’s take a look at an example that uses these identities. The problem says evaluate inverse cosine of cosine 3 pi over 4. Now because 3 pi over 4 is in the domain of the restricted cosine function the identity holds and the answer should be 3 pi over 4. But here 5 pi over 3 is not between zero and pi, so inverse cosine of cosine of x will not equal x in this case. In order to evaluate this, I’ve got to fist evaluate cosine of 5 pi over 3. Inverse cosine of, and what’s the cosine of 5 pi over 3? It’s ½. Now what’s the inverse cosine of ½? That would be the angle whose cosine is ½ as long as that angle is between 0 and pi and the answer is pi over 3. So here you see the inverse cosine of the cosine of something doesn’t have to equal that something. If the original angle’s not in the interval form zero to pi.
Finally, cosine of inverse cosine of pi, this expression doesn’t work because pi is not in the domain of inverse cosine. Remember that the domain of inverse cosine is the numbers from -1 to one so this is actually undefined. The inverse identities for inverse cosine and cosine remember that these only apply when x is in the domain of inverse cosine for this identity and when x is in the domain of the restricted cosine function for this one.