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 = 3/4 and θ terminates in quadrant III

Accepted Answer

Hello, everyone. We want to find the values of the sine of the and the cosine of the, if the cosine of two, theta equals 7 24th and theta terminates in the third quadrant. Our answer choices are a, the sine of theta equals negative square root of 39 divided by eight. And the cosine of theta equals the square root of 83 divided by eight B. The sine of theta equals the square root of 51 divided by eight. And the cosine of theta equals the square root of 93 divided by eight C. The sine of theta equals negative square root of 39 divided by 12. The cosine of theta equals negative square root of 83 divided by 12 D. The sine of theta equals negative square root of 51 divided by 12. The cosine of theta equals negative square root of 93 divided by 12. So first thing I notice is that I need to find both sine and cosine of theta. And I have double angle theorems for cosine that might help me do both. So I'm going to start with the one that'll help me find the sine of theta. And that's that the cosine of two theta has to equal one minus two sine squared theta. There's a couple different ways I can do this. I am going to put the value of cosine of two theta in and then I'm going to solve until I have the sign of theta equals a value. So the cosine of two theta is 7 24th and that equals one minus two sine squared theta. So I want to get the sign of the by itself. So the first thing I'm gonna do is subtract one from both sides. So when I subtract one from 7 24th, I'm gonna subtract it as 24 24th. She'll give me negative 17, 24th equals negative to sine squared theta. Now to get the sine squared part alone, I'm gonna divide both sides by two. I'm sorry, negative two, which is like multiplying by the reciprocal of negative one half. So I'm gonna multiply here by negative one half. And this will give me a positive 17, divided by 48 equals the S squared of theta. And I'm going to move up to the next column. So I have that the sine squared of theta equals 17 divided by 48 to get signed by itself. I'm gonna take the square root of both sides. So the sign of the will equal the square root of 17 divided by the square root of 48. And I want to simplify this and rationalize it. So the square root of 48 could be rewritten as four multiplied by the square to three. And my numerator is gonna stay the square to 17. To rationalize this. I'm multiplying numerator and denominator by the square root of three. And that will give me in my numerator, the square root of 51 in my denominator four multiplied by the square root of three times the square to three will be 12. And since we know we're in the third quadrant and sine is negative in the third quadrant, I could say the sine of theta will equal negative squared of 51 divided by 12. So we're halfway there, we know what the sine of theta is. Now we need to find the cosine of the. So luckily we have another double angle identity that will help us find that. So we have the cosine of two, theta equals two, multiplied by the cosine squared of theta minus one. And I'm going to do the same thing I did last time where I'm putting the value in right away. And then I'll work on solving for the cosine of the. So the cosine of two theta was 7 24th and that's gonna equal two cosine squared theta minus one. So in solving, I'm gonna add one to both sides and to add it to the fraction. I'm adding it as 24 24th. When I add those together, I get 31 divided by 24 equals two cosine squared theta. Next, I'm gonna divide both sides by two, which is gonna be the same as multiplying by the reciprocal of one half. So I will have 31 divided by 48 equals the cosine squared of theta. And now to get cosine by itself, I'm gonna take the square root of both sides and I'm actually gonna flip flop it a little bit just so it says cosine of theta first and that'll equal the square root of 31 divided by the square root of 48. Again, I wanna rationalize my denominator. Lucky for us, it's the same denominator it was in the previous one. So we'll have the square to 31 as our numerator. The denominator will be four, multiplied by the square root of three to rationalize it and multiplying both numerator and denominator by the square root of three. So the cosine of theta, it's gonna equal the square root of 93 divided by 12. And again, we're in quadrant three and in quadrant three cosines negative. So this will be a negative square root of 93 divided by 12. That's our cosign of theta. And now that we have both pieces, let's check our answer choices. And this is answer choice. D, the sine of theta equals negative square root of 51 divided by 12. And the cosine of theta equals negative square root of 93 divided by 12 have a nice day.

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

θ, given cos 2θ = 3/4 and θ terminates in quadrant III

Key Concepts

Double-Angle Identity for Cosine

Sign of Trigonometric Functions in Quadrants

Pythagorean Identity

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