>> The equations behind me are called sum and difference formulas or sum and difference identities and let's just kinda get the lay of the land here. They look a little complicated at first but they're really not any big deal. Notice this is the sine of the sum of angles and this is the sine of the difference of angles and they're associated with a certain expression involving sine and cosine over here. And here we have the cosine and of a sum, cosine of a difference of angles, and then an expression, tangent of a sum, tangent of a difference, expression. Okay. So that's the idea. Now these formulas or identities, we can think of them either way, are useful in both directions. That is, sometimes we'll use them in this direction and sometimes in this direction. Sometimes we'll have an expression like this and we'll rewrite like that. You see, and sometimes we'll have this and we'll rewrite it like that. You know, in using these, you'll begin to notice a pattern and the pattern will allow you to remember these rather easily and I'm not suggesting that you go through and memorize these. I am suggesting that you write these down or refer to them in your textbook when you first have to use them but be aware, be conscious of the pattern situation involved in these and with that consciousness, the memory of them will come back rather easily. For example in the sine of the sum and difference, we have sine cosine, cosine sine in both of them and we have one is plus, one is minus. That's the only difference between these two expressions and this is the expression for the sine of the sum of angles and this is the expression for the sine of difference of angles and in this pair, there are--there's a nice little pattern. Now this is the cosine of a sum and cosine of a difference but cosine cosine and sine sine, cosine cosine, sine sine, so that part of the idea is the same in both of them and when we're talking about the cosine of the sum of angles, we have a minus sine here, oddly. And when we have the cosine of the difference of angles we have a plus sine here, oddly, right, and then we have the tangent--the sum and difference of--tangent of the sum and difference of angles, similar kinds of patterns emerge. We have fractions in both cases for example. And when we have the tangent of the sum of angles, it's the tangent of one angle plus the tangent of the other angle in the numerator and then it's one minus the product in the denominator. And when we're talking about the tangent of a difference, it's the difference in tangents of the two angles and in the denominator one plus the product. Okay. So anyway, a lot of patterns emerge in this situation. Here's one--formulas or identities can be used. We can evaluate exactly the sine of, for example, 105 degrees if we rewrite 105 degrees as the sum or difference of some familiar angles. So we can rewrite this as the sine of 60 degrees plus 45 degrees, you see and then implement one of our formulas over here. Well it would be the formula associated with the sine of the sum of angles. Now--so we refer back to that formula and it's the sine of the first angle times the cosine of the second angle plus the cosine of the first angle times the sine of the second angle and then we go through the process of evaluation by just kinda using our little triangles and so forth and we would write those triangles down maybe on the side or something like that and find the sine of 60 degrees to be the square root of 3 over 2, and the cosine of 45 degrees is 1 over the square root of 2, and the cosine of 60 degrees is 1/2, and the sine of 45 degrees is 1 over the square root of 2, and now it's a matter of cleaning up algebraically. This becomes--now just multiplying the two fractions, this is the square root of 3 over 2 root 2 and here this is 1 over 2 root 2. Now from here, there are really two ways to rewrite this expression and I think it's important that we recognize those two ways. Actually, there's more than two ways. There are number of ways of rewriting this and it's important to know the difference between them because your--the answers in your text book take a particular form--you'll get the same answer on your paper but it'll just be in another algebraic form. And you're gonna think that you've made a mistake after in this part of the problem when, in fact, you just made a slight mistake maybe or not a mistake at all in the expression of your answer. It's important to be able to recognize that things that may look quite a bit different are precisely the same. As an example, in these two you recognize that we have common denominators of 2 root--