We’re talking about the inverse cosine function. And remember that the inverse cosine function is the inverse of the restricted cosine function, which is the cosine function restricted to the interval between zero and pi. We’ve graphed it already just by taking key points of the restricted cosine function and interchanging the x and the y coordinates and we got this and we drew a smooth curve. This is a graph of inverse cosine function and the idea behind the inverse cosine function is this, that x equals cosine of y.
Let’s take a look at our definition. Y equal inverse cosine of x means x equals cosine y, where y is between 0 and pi. The way to think about inverse cosine is it’s the number, in the interval between zero and pi whose cosine is x. Let me do an example. Inverse cosine of 1/2.
What number has a cosine equal to ½? There are lots of number that do but the only number between zero and pi is pi over 3. Cosine of pi over 3 equals ½ and pi over 3 is in the interval from 0 to pi. Now we would write inverse cosine of ½ equals pi over 3. Now there are other numbers that have a cosine of 1/2 like negative pi over 3. Negative pi over 3 is not in this interval so it would be the inverse cosine of 1/2. Also 5pi over 3, infinitely many. So inverse cosine will give you the one value between zero and pi that has a cosine of ½.
Let’s take a look at another example. What angle has a cosine of -1? Well pi, cosine of pi is -1. It’s certainly not the only angle that has a cosine of -1 but it is the only one on this interval and pi is in the interval from zero to pi. You don’t always s have to write this, I’m just trying to emphasize the point that the number that you choose for the value of inverse cosine’s got to be on this interval. So inverse cosine of -1 is pi. And there are infinitely many angles that have a cosine of -1 like – pi or 3 pi but only one of them lives in this interval.
Finally inverse cosine of 2. Let’s take another look at the graph of the inverse cosine function. Notice that its domain is between -1 and 1. So it’s not defined for 2. Remember that the cosine function’s only capable of outputting numbers between -1 and 1, that’s the range of cosine. The range of cosine becomes the domain of inverse cosine. So 2 is not in the domain so this is undefined.
Let’s write this down, inverse cosine x, domain between -1 and 1. Range between zero and pi. The thing to remember about inverse trig functions is that they output angles. Just as the trig functions take in inputs that are angles and output numbers, the inverse trig functions take in numbers and output angles.