 Suppose we want to find the sine or cosine of the sum of two angles. To begin with, suppose we can draw the right triangles with angles phi and theta and place them in a rectangle as shown. Now, since A, B, C is the right angle, let me know the sum of the angle CAB and ACB is 90. And so, angle ACB must be 90 minus theta. Now, since ACD is also a right angle, we know that these three angles, ACB, 90, and DCE, add to 180. And so, that says that DCE is theta. So, we want to find the sine of theta plus phi. And we know that the sine of theta plus phi is opposite. We'll draw that perpendicular and call it DF over hypotenuse AD. So, let's try to find DF. So, DF is the same length as BE, and that's CE plus CB. Now, let's consider CB that's in this right triangle. And so, we know that sine of theta is CB over AC, or CB is AC sine theta. Likewise, CE, well, that's in this right triangle. And because this is the angle theta, then the cosine of theta is CE over CD. And so, CE is CD cosine theta. So, in our equation, we can replace, but wait, there's more. We now need to know something about CD and AC. Now, CD is in this right triangle. And so, we know the sine of phi, well, that's CD over AD. And AC is in this right triangle. And so, that's the cosine of phi is AC over AD. And so, we can find AC and CD, which will be. And substituting them back gives us, and if we divide everything by AD, we get, but the ratio DF to AD, well, that's just the sine of phi plus theta. And so, we have our relationship. And this generalizes even if we don't have acute angles. And giving us a useful theorem, the sine of a sum can be found by a formula. So, for example, let's find the sine of 75 degrees. And we know that 75 degrees can be rewritten as the sum of 30 and 45. And here's the important thing. We know the sine and cosine of 30 and 45. And so, we can rewrite sine of 75 is the sine of 30 plus 45. Our theorem gives us a formula, substituting in our values, and simplifying. What about the sine of a minus b? So, we know that a minus b, well, that's really the same as a plus negative b. And so, we can treat this as the sine of a sum. But the cosine of negative b is just the cosine of b. And the sine of negative b is negative the sine of b. And so, we have... And so, we can extend our result to include the difference of angles as well. So, let's find sine of 75. Wait, didn't we just find that? Well, while we already found sine of 75, we can verify that our new method gives us the same result. As a general rule, if you have a new way of doing something, make sure it gives the same answer as the old way. So, I want to write 75 as a difference. How about 120 minus 45? And so, the sine of the difference is, substituting in our values, and simplifying. Uh-oh. Oh, wait, they are the same. We're good.