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

Question

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

θ, given cos 2θ = 2/3 and 90° < θ <180°

Accepted Answer

Hello, everyone. We are asked to find the values of sine of theta and cosine theta. If cosine to theta equals negative 8/15 and theta is between 9180 degrees. Our answer choices are a, the sine of theta equals the square root of 570. Divided by 15. The cosine of theta equals negative square root of 31 divided by 15 B. The sine of theta equals negative squared of 31 divided by 15. The cosine of theta equals the square root of 570 divided by 15 C. The sine of theta equals the square root of 690 divided by 30. The cosine of theta equals negative square root of 210 divided by 30 D. The sine of theta equals negative square root of 210 divided by 30. The cosine of theta equals the square root of 690 divided by 30. So thinking about my double angle identities for cosine to theta. I know I have one that should help me with sine and one that should help me with cosine. So I'm gonna start with the, the identity that the cosine of two theta equals one minus two sine squared theta. And I'm going to use that the cosine of two theta is negative 8/15 right off the bat. And then I'm going to solve for the sine of theta. So negative 8/15 equals one minus two sine squared theta. So to get the sign of data alone, I'm first gonna subtract one from both sides. And when I subtract one from my fraction, I'm gonna subtract it as 15 15th. And this will give me negative 23 15th equals negative two sine squared. Theta. Now I'm gonna divide both sides by negative two which with a fraction means I'm multiplying it by the reciprocal negative one half. So I will have a positive 23 divided by 30 equals sine squared theta. I'm going to bring this up to the next column to have a little bit more space. So the sine squared of theta equals 23 divided by 30. I'm gonna take the square root of both sides. So I get the sine of theta equals the square root of 23 divided by the square root of 30. And I want to rationalize my denominator. I don't think it can be reduced. So I will rationalize it by multiplying numerator and denominator by the square root of 30. This will be that the sine of theta equals the square root of 690 divided by 30. And seeing that we're in quadrant two sine would be positive. So I can say this is the sine of theta, the square root of 690 divided by 30 halfway there, we want to find now the cosine of theta. So using a different cosine double angle identity will say that the cosine of two theta equals two multiplied by the cosine squared of theta minus one. Again, I am not gonna rearrange this and then plug values in, I'm plugging the value in right away. So I have negative 8/15 equals two cosine squared of theta, right, two cosine squared of theta minus one. So the first thing to solve here, I'm gonna add one to both sides and I'm gonna add it to the fraction as 15 15th. So this will give me 7/15 equals two cosine squared of theta. And I want the cosine of the by itself. So I'm gonna divide both sides by two, which means I'm gonna multiply my fraction by the reciprocal of one half. And we will have seven divided by 30 equals the cosine squared of theta. Next, I'm gonna take the square root of both sides and I'm gonna rearrange this. So my cosine of theta is first just because I like having that first. And I will have that the square root of seven divided by the square root of 30 is our cosine of theta. But we need to rationalize our denominator. So we are going to multiply numerator and denominator by the square root of 30. And this will give us that the cosine of theta equals the square root of 210 divided by 30. But we need to consider that we are in the second quadrant and in quadrant two cosine is negative. So our cosine of theta is negative square root of 210 divided by 30. Checking my answer choices. Now that I know both the sine of theta and the cosine of theta. This matches answer choice. C where the sine of theta is the square root of 690 divided by 30. And the cosine of theta is negative square root of 210 divided by 30. Have a nice day.

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

θ, given cos 2θ = 2/3 and 90° < θ <180°

Key Concepts

Double-Angle Identity for Cosine

Quadrant and Sign of Trigonometric Functions

Pythagorean Identity

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