Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Double angle formulas are developed from our sum formulas. If we had thought about wanting sine of two Theta, we could think of that as sine Theta plus Theta and use our identity of are some formula that if we had sine of alpha plus beta that was sine alpha, cosine beta plus cosine alpha sine beta. Well, for this one, we're going to have alpha and beta be the same angle, so we're going to have sine, Theta, cosine Theta plus cosine Theta, sine, Theta. The fact that we can multiply in any order we want by the commutative property says that if sine Theta, cosine Theta plus cosine Theta, sine, Theta, it's really two of them, so two sine Theta, cosine Theta. So our sine 2 Theta formula just says that that equals two sine Theta, cosine Theta. If we wanted to look at our cosine 2 Theta formula, we would think of that as cosine Theta plus Theta. And once again remembering our cosine alpha plus beta formula was cosine alpha, cosine beta minus sine alpha, sine beta. So for this one, we're going to have cosine Theta, cosine Theta minus sine Theta, sine Theta. So cosine 2 Theta just simplifies down to cosine squared Theta minus sine squared Theta, and that's our double angle for cosine. We actually are going to have three of them for cosine because we're going to use our Pythagorean identities. If we know that cosine 2 Theta equals cosine squared Theta minus sine squared Theta instead of sine squared Theta, we could make that into one minus cosine squared Theta because we know from our Pythagorean identities that sine squared Theta plus cosine squared Theta equal 1. So if we continue this, we're going to distribute that negative and we're going to get cosine squared Theta -1 plus cosine squared Theta, or we'd get 2 cosine squared thetas -1 equaling cosine 2 Theta. So that's a second cosine double angle identity. The last one we're going to get by instead of replacing the sine, we're going to replace the cosine. So instead of cosine squared Theta to start, we're going to have one minus sine squared Theta based off that Pythagorean identity. And then it's still going to have that minus sine squared Theta at the end. So that one minus cosine or one minus sine squared Theta is all cosine squared. So our last one is cosine 2 Theta equal 1 -, 2 sine squared Theta. So the cosine double angles have three. Next we're going to look at tangent. The tangent double angle formula comes from our tangent alpha plus beta formula, which said. That's tangent alpha plus tangent beta over 1 minus tangent alpha, tangent beta. So if I have tangent two Theta, I'm going to think of that as tangent Theta plus Theta or that's going to equal tangent Theta plus tangent Theta over 1 minus tangent Theta, tangent Theta, or two tangent thetas over 1 minus tangent squared Theta. So that's our double angle for tangent. We also have some other important formulas, one being the power reducing formula. And the power reducing formula is going to get based off of the cosine formula and the double angle cosine formula. That said cosine 2 Theta equals two cosine squared Theta -1. So if I wanted to solve for the cosine squared, we would start by adding a one to the other side and dividing by two. So one plus cosine 2 Theta all divided by two equals cosine squared Theta. If we wanted to do the same thing for our sine, we had cosine 2 Theta equaling 1 -, 2 sine squared Theta. Start by subtracting the one and that's going to equal -2 sine squared Theta. So if we divide by a -2, that's going to change all the sines. So we'd have one minus cosine two Theta all over two. So those are two of our power reducing formulas. We also have a tangent one. The tangent one we can find simply by saying tangent squared Theta is really sine squared Theta over cosine squared Theta. And we just developed the sine squared Theta was one minus cosine 2 Theta over 2. Cosine squared Theta was one plus cosine 2 Theta over two. Those twos are going to cancel. And so we're going to have one minus cosine 2 Theta over one plus cosine 2 Theta. From there, we're actually going to come up with what are called half angle formulas. So we want to remember these 3 formulas, the last one being tangent squared Theta equal 1 minus cosine 2 Theta over one plus cosine 2 Theta. If we start with these three formulas, so one plus cosine 2 Theta over 2 equals cosine squared Theta and one minus cosine 2 Theta over 2 equals sine squared Theta and tangent squared Theta equals 1 minus cosine 2 Theta over one plus cosine 2 Theta. What we want to do is we want to think about changing our variables. Variables are just variables. So we're going to introduce alpha. Let's let alpha equal 2 Theta. If alpha equals two Theta, then alpha divided by two would equal Theta. We're also going to get rid of the square roots. So if we have one plus cosine instead of two Theta, we're going to call it alpha. Over here, we're going to have cosine squared instead of Theta. That's alpha over two. If I wanted to get rid of the square root or the square, I square root each side and we would remember positive and negative because we need to pay attention to which quadrant the half angle is actually occurring in. So cosine of alpha over 2 is really just positive or negative square root 1 plus cosine alpha over two. If we did the same thing for sine over here, we'd have one minus cosine alpha over 2 equaling sine squared the quantity alpha over 2. Getting rid of the square. We have this positive negative sqrt 1 minus cosine alpha over two equaling sine alpha over 2. The last one. We'd have tangent if I go ahead and get rid of that square. At the same time, I'd have tangent alpha over two equaling positive negative square root, 1 minus cosine alpha over one plus cosine alpha. Now tangent actually is going to have three different possible cases. That's one of them. The other two that are very powerful come from multiplying by its conjugate. If I have sqrt 1 minus cosine alpha over 1 minus cosine alpha, then that would equal sqrt 1 minus cosine alpha squared on the top. And if we foil out the bottom we'd have one minus cosine squared alpha. But we know that one minus cosine squared alpha is really just sine squared. So what happens with this is the square root and the squares are going to cancel and we're going to have one minus cosine alpha over sine alpha. Now with a little bit of thought, if we think about alpha and half alpha, what happens is the sines the positive and negatives of the alphas will be the same as the tangent of the alpha over 2. The third formula for tangent would be to do the conjugate but to do it of the top instead of the bottom. So starting out with the original formula of 1 minus cosine alpha over one plus cosine alpha, this time multiply by 1 plus cosine alpha over one plus cosine alpha. That gives us positive negative square root. If we foil at the top, we get one minus cosine squared alpha, which is really just sine squared alpha, and the bottom turns into one plus cosine alpha squared. So we're going to have sine squared alpha over one plus cosine alpha squared. The square roots and the squares are going to cancel, so we'd have sine alpha over one plus cosine alpha. So to summarize our three tangent formulas, tangent alpha over 2 equal positive or negative square root, one minus cosine alpha over one plus cosine alpha. For that one, we do have to pay attention to quadrants. For these other two, the quadrants already been taken into account which is very helpful. So 1 minus cosine alpha over sine alpha or sine alpha over one plus cosine alpha. Thank you and have a wonderful day.