 In our previous video, we talked about the angle sum and angle difference formulas for cosine. And as a reminder, we saw that cosine of A plus B, this was equal to cosine of A, cosine of B, minus sine of A, sine of B, because remember, cosine's a jerk. It loves the cosines, it hates the sines, all right? And then similarly, if you take cosine of A minus B, you'll end up with cosine of A times cosine of B plus this time, sine of A times sine of B, like so. So you'll notice there's a sine difference. So when you have plus versus minus, this minus then becomes a plus. So you just switch the sines as you go between the two, like so, all right? Using this, we could actually prove the sum and difference angle formulas for sine. And this actually is a consequence of the co-function theorem, which remember, the co-function theorem tells us that sine of any angle theta is just equal to, of course, cosine, excuse me, cosine of pi halves minus theta. You can always switch, you can always switch to the complementary function. We're gonna use that in this conversation right now. So consider the following equation. We're gonna prove this trigonometric identity. Sine of A plus B is equal to sine of A cosine of B plus cosine of A sine of B. So let's take a look at that. So proving this trigonometric identity, let's start with the left-hand side, sine of A plus B. Well, using the co-function theorem, sine of A plus B is the same thing as cosine of pi halves minus A plus B, like so, all right? So again, that's just the co-function theorem. I'm gonna redo the angle a little bit. We're gonna take cosine of, let's think of this way. We're gonna take cosine, you can distribute this negative sign onto the A into the B. So we're gonna get cosine of pi halves minus A and then we're gonna subtract from that B, like so. So we can make that substitution here. So then we can use the angle difference identity to help us out here because after all, we're subtracting the B. And so if we do that, we're gonna get cosine of pi halves minus A. We're then gonna get cosine of B and then we're gonna add to that sine of pi halves minus A and then we're gonna get sine of B, like so, that's really nice. But then we have these cosines of pi halves minus A. The co-function theorem goes the other way, right? Cosine of pi halves minus A, that's just a sine. And so this is gonna give us sine of A times cosine of B. And then this next one, sine of pi, excuse me, sine of pi halves minus A, by the co-function theorem, that's just a cosine. And so we end up with cosine of A times sine of B, thus proving the identity that we wanted because after all, this is now the right-hand side that we're looking for. So we get that sine of A plus B is equal to sine of A, cosine of B plus cosine of A, sine of B. So I told you the way to remember the cosine identity was that cosine was jerk, right? How dare you cosine? Because cosine prefers other cosines and doesn't like other sines. Well, it turns out that sine is much, much, much nicer, right? Sine is willing to live with everyone, kumbaya type of thing. Because notice what happens with sine of A plus B. You get a sine of A, cosine of B plus a cosine of A, sine of B, right? So the cosines and the sines, they're living together in this utopian harmony kumbaya, kumbaya, that type of stuff, right? Sine of A, cosine of B, cosine of A, sine of B. And notice there's a plus sign right there, right? There's, so even though the sines and cosines are living together, right? There's no one faction that's preferred over the other. So sine is the ideal utopian society and cosine is the dystopian society where one faction is considered better than the other. It's, again, a silly story but it'll help you remember these identities. Sine of A plus B is equal to sine of A, cosine of B plus cosine of A times sine of B, utopian, right? What about the angle difference? Well, it turns out that when you went from this angle sum with cosine to the angle difference, you just switched all the sines. We're gonna do the same thing for sine here. So notice you have sine of A plus B and in which case there's gonna be a plus, sine A, cosine B and cosine A, sine B. When you switch to the difference right here, you're just gonna switch this sine as well. So as you switch from a positive to a negative, you're gonna switch from positive to negative. And this is a very simple trigonometric identity to see as well. Take the left-hand side, which is equal to sine of A minus B. We're gonna do the same trick that we did with cosine here, we're gonna treat this as sine of A plus negative B. So every difference can be thought of as a sum in which case then you're gonna get sine of A cosine of negative B plus cosine of A times sine of negative B. We now use the symmetry identities. Cosine of negative B, since cosine's an even function, it just eats up the negative sign. You're just gonna get a sine of A cosine of B. And then for the next one, sine is an odd function so it actually spits that negative sign out and you end up with the minus cosine of A sine of B. Thus giving us the right-hand side, proving the trigonometric identity. So now we have the angle difference and angle sum identities for sine. Let's put them into practice here. In the previous video, I gave you the challenge to compute cosine of 15 degrees. All right, which in radians, just so you know cosine of 15 degrees is the same thing as pi twelfths. I asked you to do that using the angle difference identity. But by the co-function theorem, cosine of 15 degrees is actually the same as sine of 75 degrees, which again, if you wanted to put that into radians, this would be sine of five pi twelfths, like so. So we can use the angle sum identity to compute the sine of 75 degrees for sine or we can use the angle difference identity for cosine to do cosine of 15 degrees. So check your answer with what hopefully you did with what we're gonna do right now. So sine of 75 degrees, I can treat this as an angle sum because 75 degrees is the same thing as 45 degrees plus 30 degrees, like so. And using the angle sum identity we had previously, this will be sine of 45 degrees times cosine of 30 degrees. Then we're gonna get cosine of 45 degrees times sine of 30 degrees. Notice that as I wrote this out, I went in the same order that I had here, 45 and 30. So just keep the same order of the angles and you're gonna toggle between sine and cosine. So here was a sine, there's a cosine. Here's a cosine, there's a sine. And make sure each product has both a sine and a cosine. You're gonna be just fine there. All right, now sine of 45 degrees is root two over two. Cosine of 30 degrees is root three over two. Cosine of 45 degrees is also root two over two. And sine of 30 degrees is one half. So this will look like the square root of six over four plus the square root of two over four. Put that together, we get the square root of six plus the square root of two over four. So this is sine of 75 degrees. This is also cosine of 15 degrees. Let me summarize what we've now observed so far. So we've now have discovered that sine of 15 degrees which is the same thing as cosine of 75 degrees. We got that to be the square root of six minus square root of two over four. We found that on the previous video. And then we now also have that sine of 75 degrees is equal to cosine of 15 degrees which is equal to the square root of six plus the square root of two over four. Which you'll notice that of course by putting a positive here, this is bigger than the other one, which is not surprising. Sine of 75 should be bigger than sine of 15. Sine of 15 should be close to zero. Sine of 75 should be close to one. Cosine of course does the exact opposite there.