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.

2θ, given cos θ = (√3)/5 and sin θ > 0

Accepted Answer

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 equals the square root of 15, divided by 17 and the sine of theta is greater than zero. Our answer choices are a, the sine of two theta equals two, multiplied by the square root of 4110. All of which is divided by 289. The cosine of two theta equals negative 259 divided by 289. B. The sine of two theta equals two, multiplied by the square root of 2191. All of which is divided by 289. The cosine of two theta equals negative 259 divided by 289 C. The sine of two theta equals two multiplied by the square root of 1187. All of which divided by 289. The cosine of two theta equals negative 134 divided by 289 D. The sine of two theta equals the square root of 161 divided by 15 and the cosine of two theta equals negative seven divided by 15. OK. So where do I wanna begin? I know the cosine of theta. And I recall that the formula for the cosine of two theta is that the cosine of two, theta equals two, multiplied by the cosine squared of theta minus one. Since I know the cosine of theta is the square root of 15 divided by 17. I'm gonna start here. So the cosine of two theta, I'm going to plug in squared of 15 divided by 17. So I have two multiplied by the square of 15 divided by 17 squared minus one. I'm going to do out the square. So I have two multiplied now by the square to 15 squared is 15, 17 squared is 289 minus one. Next, I'm gonna multiply the two into the parentheses. So two multiplied by 15, divided by 289 will give me 30 divided by 289 minus one. I'm gonna treat the one as 289 divided by 289 and do the subtraction. I get the cosine of two theta equals negative 259 divided by 289. So I have half my answer. I know that the cosine of two theta is negative 259 divided by 289. Next, I'm gonna find the sign of too the So the formula for the S of two, theta is two multiplied by the sine of theta cosine of theta. Well, unfortunately, we don't know the sign of the yet, but we can find it, we will use the Pythagorean theorem. It says X squared plus Y squared equals R squared. And our knowledge that the cosine of theta is the ratio of X divided by R. And recall that the sine of theta will be Y divided by R. And knowing that the cosine of the is the square root of 15 divided by 17. I could say X is the square to 15. Why is unknown R is 17? And now I'm going to use the Pythagorean theorem to solve for Y. So I can find the sign of the. So we have X squared. So the square root of 15 squared plus Y squared equals R squared, R is 17 squared. The square to 15 squared is 15 plus Y squared equals 17 squared, which is 289. I wanna subtract 15 from both sides, Y squared equals 274 to undo the square. I'm gonna take the square root of both sides. Y equals the square root of 274. Since we know Sine is greater than zero, we only need to take into consideration a positive for the square root. And we'll find that the sign of theta is our Y value we just found. So the square root of 274 divided by R which is 17. So now that we know that the sign of the is the square root of 274 divided by 17, we can plug both that and the cosine of the into the double angle formula for sine. So the sign of two theta, it is going to equal two, multiplied by the square root of 274 divided by 17, multiplied by the cosine of theta, which was the square root of 15, divided by 17. I'm going to multiply straight across the top. I'm gonna leave the two out front for now. I think I'm gonna deal with the fractions first. So the square root of 274 multiplied by the square root of 15. We can keep the square root and multiply the values underneath. And that will give us the square root of 4110. And my denominator 17 multiplied by 17 is 289. The only other thing I could do here is I can say I'm multiplying the two by the numerator. So I'd get the sine of two, theta equals two multiplied by the square root of 4110 all of which is divided by 289. That can't be simplified any further. So that's what the sign of two theta will equal. So comparing the sine of too the and the cosine of tooth, the, that we found this appears to be answer choice. A where the sine of two theta equals two, multiplied by the square root of 4110 all of which is divided by 289. And the cosine of two theta equals negative 259 divided by 289 have a nice day.

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

2θ, given cos θ = (√3)/5 and sin θ > 0

Key Concepts

Double-Angle Formulas

Pythagorean Identity

Quadrant and Sign of Trigonometric Functions

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