 What about the cosine of a sum or difference? So, as before, let's assume we can draw right triangles, angles phi and theta, and place them in the rectangle as shown. And, again, as before, angle DCE also has measured theta. Since the cosine of phi plus theta is the ratio af to ad, we want to find af. So we see that af is ab minus fb, and ab is in this right triangle, and so we see that the cosine of theta is ab over ac, and so ab is a cosine theta. To find fb, we note that it's the same as de, which is in this right triangle, and so sine of theta is de over cd, and so de is cd sine theta. Now, we also need ac and cd, and they're both in this right triangle, and so cosine phi ac over ad, so ac is ad cosine phi, and sine of phi cd over ad, and so cd is ad sine phi, divide by ad, and since the cosine of phi plus theta is af over ad, we have our relationship, which suggests, so we might find cosine of 105 degrees, and so we have our relationship, and we note that 105 is 60 plus 45, and we know the sine and cosine of 60 and 45, so we can rewrite, evaluate, and simplify. For a difference, we'll note that the cosine of a minus b, well, that's the same as the cosine of a plus negative b, so now this is the cosine of a sum, since cosine of negative b is cosine of b, and sine of negative b is negative sine of b, we can simplify, and so we can extend our result to include the cosine of a difference. Well, let's try to find cosine of 105 degrees, this time using a difference. We observe that 105 is 180 minus 75, and so we can use our formula, and we need cosine of 75 and sine of 75, but we don't know these values, so we should try a difference of traction. So we might try 135 minus 30, and this time we do know the sine and cosine of both 135 and 30, so we can evaluate and simplify, and we get the same answer as we got before, this time using a difference.