 The angle sum identities allow us to find the sine or cosine of any angle we can write as a sum or difference of angles of known sines or cosines. Since we know the sine and cosine at 30 degrees, 45 degrees, and 60 degrees, we can find the sine or cosine of 75 degrees, 15 degrees, and others. But we can also find these using a calculator, so what if we don't know the angle measures? For example, suppose A is a second quadrant angle where the sine is one-third, let's find the sine of A plus 45 degrees. So our angle sum formula would tell us that sine of A plus 45 will be... Now we know the sine and cosine of 45 degrees, and we're given the sine of A, but we need the cosine of A. And we can use the Pythagorean identity and find... Now we do have to make a choice of whether to use the positive or negative square root. Since A is a second quadrant angle, the cosine of A should be negative, so we'll take the negative square root and simplify it a little bit, then substitute back, and find... Or suppose we have an angle in the first quadrant where we know the cosine. Let's find the sine of twice the angle, the cosine of twice the angle, and then determine what quadrant twice the angle will be in. So we're given the cosine, we do need to find the sine. So we'll use the Pythagorean identity to find the sine. And since A is an angle in the first quadrant, we'll take the positive square root. And note that it will be convenient to simplify the denominator, but not necessarily the numerator. Now, since 2A is A plus A, then we can use our angle sum formula for cosine to find. Now we could find sine of 2A using the Pythagorean identity, but we won't. Let's use the angle sum formula for practice. So sine of 2A is the sine of A plus A, which will be... Now since cosine 2A is negative and sine 2A is positive, then 2A must be an angle in the second quadrant. Now there's one other advantage to this abstract reasoning. It can help you remember the formulas. So remember, the answer should be the same no matter how you find it. For example, suppose you don't quite remember the angle sum formula for sine, and you're pretty sure it's either this one or it's this one. Since sine of A plus 0 is A, then applying the correct angle addition formula to sine of A plus 0 should give you the sine of A. So maybe the sine of the sum is this first formula. Let's try it out. Sine of A plus 0 will be... And remember, the sine of 0 is 0 and the cosine of 0 is 1. So the right hand side simplifies to... And I don't believe it. So it's not this formula. Well, if it's not the first thing you think of, it must be the second thing you think of. Well, let's check it out. Maybe the sine of A plus B is this formula. So let's try it out. Find sine of A plus 0 and simplify. And we do get sine of A on the right hand side, so the correct formula is probably our second formula. Now, you should be careful with this approach. It'll eliminate the wrong formula, but it won't guarantee what the right formula is. So if you thought the correct formula was sine of A plus sine B, we could try it out. And it seems to work, but it is in fact not true.