 Let's try to solve the equation sine of 2 theta plus the square root of 2 times cosine of theta equals 0 and let's solve it on the domain where theta ranges from 0 degrees to 360 degrees. So this indicator right here at the end tells us we want to solve this equation in terms of degrees. So when you look at this equation, there's a mismatch of angles here. There's a sine of 2 theta and there's also a sine of theta. If we want to solve this thing effectively, we have to get the same angle and our goal is going to here be theta. Now the easiest way to accomplish that is just to look at sine of 2 theta here. We recognize we could apply the double angle identity for sine. If we do that, remember sine of 2 theta is equal to 2 sine theta times cosine theta plus square root of 2 times cosine of theta, this equals 0. So notice now the angles in play are only theta. So let's try to factor the left hand side if that's possible. Looking at the left hand side, I do see there's a common factor of cosine in both terms. Let's factor that out. That leaves, once we take the cosine theta out, that leaves behind 2 sine theta plus the square root of 2. Like so. That's a 0 equal to not theta there. Actually, I mean, we could have factored out a square root of 2 if we wanted to right here. But I think we'll just leave it alone. It'll be just fine because now we have a product of two things that equal to 0. So consider the first one, for example, cosine theta equals 0. When does that happen? In theta, the thing of the unit circle is, sorry, cosine of theta equals 0 when you're at the top and at the bottom of the unit circle. So theta would equal 90 degrees and 270 degrees. That's when cosine is equal to 0. Well, what about the other one? This one takes a little bit more work, but it won't be too arduous here. 2 sine theta plus the square root of 2 equals 0. Subtract the square root of 2. We get 2 sine theta equals negative root 2. Divide by 2, we get sine theta equals negative root 2 over 2. So thinking about the reference angle here for a moment, sine of theta equals the positive one, right, root 2 over 2. That happens when theta equals 45 degrees in the first quadrant. Now, coming back to the quadrant we're in, if we have a negative sign, that means we're going to be the third quadrant and in the fourth quadrant. So in that setting, we see that theta should equal angles at reference 45 degrees, but of course, in the third and fourth quadrant. So in the third quadrant, you take 180 plus 45, that's 225 degrees. And then in the fourth quadrant, you'll take 360 minus 45 degrees, or if you prefer 270 plus 45 degrees, you end up with 315 degrees, right? So like so. So put these together. We think that the solutions are going to be theta equals 90 degrees, 225, 270 and then 315. Do make sure you put the little degree symbol there because without such our conventions, we're supposed to assume that means radians, which would be incorrect in this situation. So we can factor, use factoring to solve these geometric equations, but we might have to use an identity. Like in this situation, the double angle identity was appropriate. Now for sine, sine of 2 theta is pretty easy. You just replace sine of 2 theta with 2 sine theta, cosine theta, and then try to factor out a common divisor there. When it comes to the double angle for cosine, there's a little bit more flexibility because there's a few more options. Consider the example, cosine of 2 theta plus 3 sine theta minus 2 equals 0. And again, let's solve it on the domain 0 to 360 degrees. Now, when it comes to cosine of 2 theta, we've learned previously, there were three different versions of the double angle identity there. There was cosine squared theta minus sine squared theta. We had 2 cosine squared theta minus 1 and we had 1 minus 2 sine squared theta. We had three different versions of the cosine of 2 theta here. Which one would be most profitable here? Well, if you think of the first one, right, we would place cosine 2 theta with a cosine squared minus a sine squared there. It's like, well, I already have a sine theta in place. The sine squared is not such a big deal, but cosine squared. What do you do? What do you do with that? We have to address that. Same thing with the second one here. Since we have a sine theta, having a cosine squared isn't so helpful for us. But on the other hand, since we have a sine theta, if we had a sine squared, that actually would be fantastic because this would create a quadratic equation in terms of sine. So when it comes to cosine of 2 theta, you're going to pick the version of cosine of 2 theta that basically makes the quadratic equation you need. In this case, it would be we want to sign because we already had a sign here. If instead this was a cosine of theta, we actually would choose this one instead. But that's not the reality. Since we have a sine theta, we're going to move forward with that. So cosine of 2 theta becomes 1 minus 2 sine squared theta. We then get plus 3 sine theta minus 2 equals 0. Combine some like terms, which basically is just the 1 and the negative 2 right there. So we get negative 2 sine squared theta plus 3 sine theta. And then we're going to get a negative 1 equals 0. I really don't like the leading coefficient to be negative. So I'm going to assign everything by negative 1. So I get 2 sine squared theta minus 3 sine theta plus 1 equals 0. And the reason I did that is when the leading coefficient is negative and I have to try to factor this thing, I feel like I'm more likely to make a mistake. It's more error-prone. So I'm just going to clean it up and have a positive coefficient in that situation. So let's think of this thing as possible to factor. So we have 2 times 1 is 2 factors of 2 that add to negative 3. That would just be negative 2, negative 1. Factorization is possible. We can go through the detailed reverse foil process. I figured since my numbers are 2 and 1, there's not a lot of divisors there. I could probably guess it. The only way to get a 2 sine square is to have a 2 sine and a sine. The only way to get a plus 1 is you're going to have a 1 and 1. It's either both positive 1 or both negative 1. But in order to make a negative 3, we're going to have to have some negative signs there, right? So you get a 2, negative 2 sine and a negative 1 sine. That gives us the negative 3 signs. So there we have it. Then now that we've had it correctly factored, we set each and every one of these factors equal to 0 and solve. So we have 2 sine theta minus 1 equals 0. That means 2 sine theta equals 1, which means sine theta equals 1 half. So I added 1 and then divided by 2 on both sides. When is sine theta equal to 1 half? Well, of course, that'll happen in the first quadrant, but also in the second quadrant here, right? So in the first quadrant, we're trying to find solutions with respect to degrees. In the first quadrant, that happens at 30 degrees, of course. In the second quadrant, we want the angle that references 30 degrees. So 180 take away 30, which is 150 degrees. And then for the second possibility, sine theta minus 1 equals 0. Sine theta minus 1 equals 0. That means sine theta equals 1. When does sine theta equal 1? When does the y coordinate on the unit circle become a 1? That happens at the top of the circle, which is 90 degrees. And so we get that solution there. Putting these together, we see that the solutions to this trigonometric equation between 0 degrees and 360 degrees would be 30 degrees, 90 degrees, and 150 degrees, like so. And so we saw some examples here of using the double angle identity to help us solve trigonometric equations by factoring. The key takeaway here is when you have the cosine of 2 theta, use the identity that will create for you a quadratic polynomial, either in terms of sine or in terms of cosine.