Hey, everyone. In this problem, we're asked to evaluate the expression that the secant of the inverse cosine of negative root 3 over 2. Now, this function might look a little bit complicated to you, but we can still break this down and solve this the way that we already know how. So let's start with that inside function here, the inverse cosine of negative root 3 over 2. Now because this is an inverse trig function, remember that we can also think of this as, okay, the cosine of what angle will give me negative root 3 over 2?
Now also remember, because this is an inverse trig function, we're looking for an angle within a specified interval. Now for the inverse cosine, I only want to work with angles that are between 0 and pi. So I am looking for an angle between 0 and pi for which my cosine is equal to negative root 3 over 2. Now looking at my unit circle, I know that for 5 pi over 6, that will give me a cosine of negative root 3 over 2. So that tells me that this inside function is 5 pi over 6, and I am now left to find the secant of that 5 pi over 6.
Now from working with these trig functions previously, you may remember that the secant is really just one over the cosine. So this secant of 5 pi over 6 can really be rewritten as one over the cosine of 5 pi over 6. Now we actually just saw what the cosine of 5 pi over 6 was. We know that it's negative root 3 over 2. So this is really 1 - 3 2 .
Now whenever I have one over a fraction, I can really just flip that fraction. So this is going to give me -2 3 . Now this is technically a correct answer, but whenever we have a radical in our denominator, we really don't want that there. So we actually want to rationalize this denominator, which we can do by multiplying by 3 3 . Now that will give me a final answer of - 2 √ 3 3 .
And that's my final answer. The secant of the inverse cosine of negative root 3 over 2 is negative 2 root 3 over 3. Thanks for watching. And let me know if you have questions.