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

2x, given tan x = 5/3  and sin x < 0

Video transcript

Hello, everyone. We are asked to find the values of the sine of two theta and cosine of two theta. If the tangent of theta equals 25 divided by seven and the sine of theta is less than zero. Our answer choices are a, the sine of two, theta equals 675 divided by 677. The cosine of two theta equals 52 divided by 677 B. The sine of two, theta equals 175 divided by 337. The cosine of two theta equals negative 288 divided by 377. See the sign of too, the equals 675 divided by 677. The cosine of two, theta equals negative 52 divided by 677 D. The sine of two theta equals 117 divided by 337. The cosine of two theta equals negative 91 divided by 337. First, I am recognizing that if we have the tangent and it is positive and we are told that we have a sign of the, that is negative that we are working in quadrant three. So I'm gonna just make a note that I'm in quadrant three and that both a sign of the and cosign of theta will ultimately be negative. So less than zero. First, I notice I have the tangent of theta. And I recall that both formulas the sine of two theta and the cosine of two theta need me to know the sine of the and the cosine of the. So I'm gonna use the tangent of theta being equal to 25 divided by seven. And I recall that tangent is the ratio of the Y value divided by the X value when we're looking at trig on a coordinate plane. So if that's the case, I could think of the cosine of theta as X divided by R and the sign of theta as Y divided by R. So I need to find out what is R, I can use the Pythagorean theorem to find this out. So X squared plus Y squared will equal R squared. Here we know X and Y X is going to be seven, we know this from the denominator of the tangent. So seven squared plus Y squared. So 25 squared equals R squared doing this out, I get 49 plus 625 equals R squared. I'm going to total this and I will get 674 equals R squared and I will take the square root of both sides and find that R equals the square root of 674. So knowing this, I can decide what my cosine of theta and my sin of the are. So the cosine of theta is the X value. So seven divided by the R value, the square root of 674. And since we know we're in quadrant three, we know this is negative. So I'm gonna write this back up at the top that the cosine of theta equals negative seven divided by the square root of 674. For the sine of theta, I need the Y value which is 25 divided by the R value, which is the square root of 674. And again, I know I'm in quadrant three and sin is negative. So our sign of the is negative 25 divided by the square root of 674. Now that I know the cosine of theta and the sine of theta, I can use the formulas for the double angles. I'm going to leave the radical in the denominator because it might come out on its own as I work with the problem. If it's still there at the end, I'll rationalize it. So I'm gonna start with the sign of Tua and that formula is two multiplied by the sine of theta cosine of theta. So we will have two multiplied by negative 25 divided by the square root of 674 multiplied by the cosine theta of negative seven divided by the square root of 674. Multiplying I'm gonna do the fractions first. So two stays on the outside. And then I'm opening a set of parentheses. Negative 25 multiplied by negative seven will give us a positive 175. Our denominator of 674 um will come from multiplying the square root of 674 by itself. And now I'm going to multiply the two into the numerator. And this will give me 350 divided by 674. And we're almost done. But I'm noticing that both of these numbers are even. So I know this can be reduced at the very least, it can be reduced by two. So if I reduce the numerator by two, I get 75 and reducing the denominator by two, I get 337. And it does not appear that there's anything else that can be reduced there. So we know the sign of tooth, the is 175 divided by 337. So we know half of the answer to this problem. We still need to find the cosine of two theta. So our formula for the cosine of two theta is that it equals two multiplied by the cosine squared of theta minus one. All right. So let's plug in what we know. So two multiplied by the cosine squared of theta. So the cosine of theta was negative seven divided by the square root of 674. So we're going to square that subtract one. I'm gonna do the square first. So two multiplied by negative seven squared is 49. The square root of 674 squared is 674 close parenthesis minus one, multiplying the two into the parentheses. I get 98 divided by 674 minus one which I'm gonna think of as 674 divided by 674. When I do that subtraction, I get negative 576 divided by 674. And I think I can reduce this fraction at the very least by two. And when I do so I get negative 288 divided by 337. So that is the cosine of two theta. So checking my answer choices looks like we have answer choice B where the sine of two theta equals 175 divided by 337. And the cosine of two theta equals negative 288 divided by 337. Have a nice day.