Angle addition identities
We've got triangle ABC here. It looks like a right triangle. We can verify that it is because it satisfies the Pythagorean theorem. 8 squared is 64 plus 15 squared is 225. 64 plus 225 is 289, which is 17 squared. So it satisfies the Pythagorean theorem. So we know that this right over here is a right triangle. But what they're asking us is-- what is the cosine of angle ABC, which is this angle right over here plus 60 degrees. Now, just like this, I don't have any clear way of evaluating it. But we do have some trig identities in our toolkit that we could use to express this in a way that we might be able to evaluate it. In particular, we know that the cosine of A plus B is equal to the cosine of A times the cosine of B minus the sine of A times the sine of B. So we could do the exact same thing here with the cosine of angle ABC plus 60 degrees. This is going to be equal to-- and let me just write out the whole thing-- the cosine of angle ABC times the cosine of 60 degrees minus the sine of angle ABC times the sine of 60 degrees. So this right over here is angle ABC. This is angle ABC. And this is 60 degrees, and this is 60 degrees. So let's evaluate each of these things. What is the cosine of angle ABC going to be equal to? I'll do that over here. The cosine of angle ABC. Well, we could go back to Sohcahtoa. Let me write it down. Cosine is defined as the adjacent-- for this angle-- the adjacent over the hypotenuse. So cosine of angle ABC is equal to 15 over 17. So this right over here is 15 over 17. What is the cosine of 60 degrees? Well, for there, we have to turn back to our knowledge of 30, 60, 90 triangles. So if I have a triangle like this-- so let me do my best here-- to construct a 30, 60, 90 right triangle. So that's a 60-degree side. This is a 30-degree side. We know-- and we've seen this multiple times-- if the hypotenuse has length 1, the side opposite the 30-degree side is one half. And then, the side opposite the 60-degree side is square root of 3 times this. So it's square root of 3 over 2. So the cosine of 60 degrees-- if you're looking at this side right over here. And let me write it in a color I haven't used yet. So I care about the cosine of 60 degrees. The cosine of 60 degrees is going to be equal to, once again, adjacent over hypotenuse. One half over 1. it's going to be equal to one half over 1, which is equal to one half. Now, let's think about the sine. Now, let's think about the sine of angle ABC. So that's this one right over here. So we have our triangle right here. Sine is opposite over hypotenuse. So opposite has length 8. Over hypotenuse, is 17. So it's equal to 8 over 17. And then, finally, we need to figure out what the sign of 60 degrees is. Well, for the 60-degree side of this right triangle, opposite over hypotenuse. So square root of 3 over 2 over 1, which is just the square root of 3 over 2. So we have all of the information we need to evaluate it. So this was the sine of 60 degrees. This whole thing is going to evaluate to cosine of angle ABC is 15 over 17 times cosine of 60 degrees is one half. So times one half. And then, we're going to subtract sine of ABC, which is 8 over 17. And then, times sine of 60, which is square root of 3 over 2. So times the square root of 3 over 2. And now, we just have to simplify it a little bit. So this is going to be equal to-- if I multiply one half times this, let's see. It's going to be 15 over 34 minus-- and let's see. We're dividing by 2. So it's 4/17ths. I'll write this 4 square roots of 3 over 17. And that's about as simple as I could do it. If I wanted, I could have a common denominator here of 34. And then, I could add the 2, so it could be 8 squared. So 3 over 34, but that still won't simplify it that much. So this is a reasonable good answer for what they are asking for. 15 over 34 minus 4 square roots of 3 over 17.