Find values of the sine and cosine functions for each angle measure.

2θ, given cos θ = -12/13 and sin θ > 0

Video transcript

Hello, everyone. We are asked to find the values of the sine of two theta and the cosine of two theta. If the cosine of theta is negative 35 37th and the sine of theta is greater than zero, our answer choices are a, the sine of two theta equals 840 divided by 1369. The cosine of two theta equals negative 1081 divided by 1369 B. The sine of two theta equals negative 840 divided by 1369. The cosine of two theta equals 1081 divided by 1369 C. The sine of two theta equals negative 24 divided by 35. The cosine of two theta equals 19 divided by 35 D. The sine of two theta equals 24 divided by 35. The cosine of two theta equals negative 19 divided by 35. So first, I know that my formula for the sign of two theta requires me to know what the sign of theta is first. So I wanna find what is the sign of theta to do that, I'm gonna use a Pythagorean identity, the identity that sine squared theta plus cosine squared, theta equals one. I am going to solve this. So it equals sine of theta. So I'm subtracting the cosine squared of theta from both sides and taking the square root. So I get that the sine of theta equals the square root of one minus the cosine squared of theta. And since we know cosine is negative, but sine of theta is greater than zero, we know we're working in quadrant two. So we only need to take into consideration the positive half of this square root. So the sine of theta is gonna equal the square root of one minus our cosine of theta which was negative 35 divided by 37 squared. So we have the sign of theta equals the square root of one minus 125 divided by one. I'm sorry, 1225 divided by 1369. Simplifying what's under the radical, I get the sign of theta equals the square root of 144 divided by 1369. And if I take the square root of both the numerator and denominator there, I find that the sine of theta equals 1237. So 12 divided by 37. So I'm gonna put that up here at the top right next to my cosine of theta. So I have those values together. So I know that the sign of theta is 12 divided by 37. So now I'm gonna find the sign of two theta. So the sign of two, theta, our formula for that is it is two multiplied by the sine of theta cosine of theta. So we're multiplying all of these numbers together. So we'll have two multiplied by our sin of theta was 12 divided by 37. Our cosine of theta is negative 35 divided by 37. So we need to multiply that all together. And when we do, we get the numerator is negative 840 the denominator is 1369. So we know that the sign of two theta equals negative 840 divided by 1369. So there's half of our answer. We still need to go ahead and find what is the cosine of two theta. So our formula for the cosine of two, theta is cosine to theta equals two, multiplied by the cosine squared of theta minus one. So we will do two multiplied by our cosine of theta was negative 35 divided by 37 squared and then minus one. So we have two multiplied by 1225 divided by 37. Oops, I'm sorry, not by 37. I have to square that also by 1369. And then we're subtracting one from there during the multiplication, I get 2450 divided by 1369 minus one. I'm going to treat the one as if it's 1369 divided by 1369. And when I do that and subtract, I get 1081 divided by 1369. And I don't think that can be reduced. I wish it could. Um But our cosign of two theta, it's gonna be 1081 divided by 1369. So this is the other part of our answer. So let's see which answer. Choice matches both of our numbers. And that appears to be answer choice. B where the sine of two theta is negative 840 divided by 1369. And the cosine of two theta equals 1081 divided by 1369. Have a nice day.