 So this is an overview of a couple of the important other formulas you should have remembered from earlier courses on trigonometry. And the one we'll focus on right now is there are two important angle addition formulas for finding the sine of cosine of an angle given that the angle is the sum of two other angles. And the angle addition formula for sine, sine of a sum or difference of two angles, sine of the first, cosine of the second, plus or minus cosine of the second, sine of the first. And here the plus or minus here inside is the same as the plus or minus here in the formula. And the cosine formula, this is the cosine of a sum or difference of two angles, that's cosine cosine minus or plus sine sine. So here if we have a sum as the angles here, we have a difference of the two expressions over there. So for example we can use these angle addition formulas to find the sine of pi minus x. Now you should actually know that the sine of pi minus x is, well this is one of those basic trigonometric ideas, the sine of pi minus x should actually just be the sine of x. And incidentally the cosine of pi minus x should be minus the cosine of x. And these are identities that you should actually know and they emerge from the geometry. But the reason that this is useful is that this is a way of remembering the details of the formulas. Everybody remembers what the formulas are vaguely, but sometimes that plus or minus can be a little bit confusing, trying to remember when you do adding and when you do subtracting. And things like that may be a little bit hard to remember. So here's a good way of remembering them. Take a situation where we already know what the answer is and make sure that our formula gives us the same answer. So I know that the sine of pi minus x, well that is the sine of a difference of two angles, that's going to be the sine, maybe it's minus, or do I do sine cosine and I don't remember which one it is. So what we can do is we can remember what the angle addition formula is, because we already know what the answer is supposed to be. If the formula doesn't give us the right answer, we know that formula is going to be wrong. So let's take a look at the sine pi, sine x, cosine pi, cosine x, well that's going to equal negative cosine x, sine of pi is zero, so this first term is gone, cosine of pi is negative one, so that gives us negative cosine x, and I know that's the wrong answer, so this first formula is not correct. Second formula, sine pi is zero, first term's gone, cosine pi is negative one, that's minus negative one cosine, that's going to give me plus cosine of x, and again I know this formula is wrong, so that formula also is not correct. And let's see, so we have sine pi cosine, sine of pi again zero, this term is gone, cosine pi is negative one, this is minus sine of x, and again that's the wrong answer. So I know that my correct formula, sine pi cosine x, cosine pi sine x minus, and that's going to give me sine of x, again sine pi is zero, cosine pi negative one, negative negative one, that's sine of x, and this is the correct answer, and what this actually tells us is that the sine of a difference, sine first cosine minus cosine times sine. Well let's actually use these to derive two other formulas that are occasionally useful, those are our double angle formulas, sine and the cosine of 2x. So let's see, sine of 2x, well that's the sine of x plus x, and that's the sine of a sum, so I can apply the sine addition formula, sine cosine plus cosine sine, and here we happen to be adding two terms that are actually the same thing, so I actually get 2 sine x cosine x. Likewise if I take cosine of 2x, well that's going to be the cosine of x plus x, and I'll apply the cosine addition formula, and that's going to be cosine cosine sine sine, and the plus changes to a minus, so that's going to be cosine x cosine x minus sine x sine x, and after all the dust settles that's cosine squared minus sine squared, and because I can express the sine of 2x in terms of the sine and cosine of x, and likewise the cosine of 2x in terms of the sine and cosine of x, these are called the double angle formulas. Generally speaking they're not really worth committing to memory, because anytime we actually need them we can derive them, and it's a useful way of remembering what the what the angle addition formulas are, but we do run into them often enough that after a while after using them a couple of times you'll remember sine 2x, 2 sine x, cosine x, and so on. Well how about half-angle formulas? So let's see if we can find the sine and cosine of x over 2, and here's a general strategy in mathematics, which is if we want to find something, what we can do is we can write down an equation that includes what we're looking for, and then solve the equation for what we want. So I want to write down an equation that involves sine of x over 2, and see if I can solve it for sine of x over 2. So our first inclination, we just derived the double angle formula, and it's useful to note x is x over 2 plus x over 2, and so I can use the sine of a sum, and that's going to be 2 sine cosine, and I can solve this equation for sine of x over 2, over on the left hand side I have sine of x, over on the right hand side 2 sine cosine, and I can try to solve this for sine of x over 2, but the problem is it involves this cosine of x over 2, and while I can use my Pythagorean identity to eliminate the cosine of x over 2, after all the dust settles, what I'm going to end up with is an equation that is cubic in sine of x over 2, and that's too hard to solve. While we can solve it, it's a little bit more challenging than we really want to deal with right now. So, well there's another formula that involves sine of x over 2, and that's going to be our cosine formula. So over on the left hand side we have cosine of x, over on the right hand side we have our sine of x over 2, and we do have this cosine of x over 2 squared, and we need to get rid of it in order to be able to solve for sine of x over 2. Fortunately, we could always fall back on our Pythagorean identity, cosine squared plus sine squared equals 1, so cosine squared is 1 minus, and here's our sine of x over 2 once again. So that means our cosine of x equals this according to our double angle formula. I can eliminate that cosine squared, and I can do a little bit of algebra, do a little bit of algebra, do a little bit of algebra, and do a little bit of algebra, sine of x over 2 is plus or minus square root 1 minus cosine over 2. Now, there is that plus or minus that I have to deal with any time I take a square root, and I have to determine that sine plus or minus s i g n sine, where I can determine whether I'm using plus or minus depending on the value of x over 2. What quadrant I'm in, the sine of that will be positive or negative, and so I do have to do that little last bit of analysis. Likewise, for if I want to find the cosine of x over 2, I can actually start with the same equation. This time I have sine squared equals 1 minus cosine squared x over 2, and again I have cosine equal cosine squared minus sine squared. I substitute sine squared 1 minus cosine squared x over 2. I do a little bit of algebra, I do a little bit of algebra, I do a little bit of algebra, I do a little bit of algebra, and I take the square root, and again I have cosine of x over 2 again plus or minus because I took the square root, and so I have to consider which I'm going to be using depending on the s i g n sine of cosine of x over 2.