Use the given information to find the exact value of each of t...

Question

Use the given information to find the exact value of each of the following: $cosine 2 theta$

$\sin θ = \frac{12}{13},$

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression using the provided data. The expression we want the value of is the cosine of two beta where the sign of beta equals 9 40 firsts and beta lies in quadrant two. We are given four answer choices. First recall that the double angle identity for the cosine of two beta would be cosine squared beta minus sine squared beta. Also notice we are given the sign of beta is nine divided by 41. So before we can use this identity, we need to find the cosine of beta. Well, luckily we know that the ratio for the sign of beta would be the value of Y divided by the value of R and that the cosine of beta would be the ratio of X to R. Right now, we don't know the value for X, but we know Y equals nine and R equals 41. And lucky for us this is based on the Pythagorean theorem. So we're gonna use the X squared plus Y squared equals R squared and the values we do know to find the value for X. And once we know X, we can know the cosine of beta. And then once we know both sine and cosine of beta, we can use our identity. So X squared plus Y squared, which is nine squared has to equal R squared, which will be 41 squared. So X squared plus 81 equals 1681. Get the X by itself, we subtract 81 from both sides. So X squared equals 1600. Recall that the opposite of a square is a square root. So to get the X by itself, we take the square root of both sides and X equals plus or minus 40. Now how do we know if it's plus or minus? Well, we are told that beta lies in quadrant two. If you think about a coordinate plane in quadrant two X is negative because it's to the left of the Y axis. So X in this case must be negative 40. Now that we know X, we can go back and say that the cosine of beta is negative 40 divided by 41. And now that we know both sine and cosine of beta, we can go back to our double angle identity. It says the cosine of two beta is gonna equal the cosine squared of beta. So we're gonna take our value of the cosine of beta negative 40 divided by 41 square it and then we're subtracting sine squared beta. So our value for the sine of beta is nine divided by 41. And we're gonna square that also, this will give us 1600 divided by 1681 minus 81 divided by 1681. And once we do that subtraction, we get 1519 divided by 1681. And this is the cosine of two beta and our answer. So our answer is 1519 divided by 1681 which is answer choice B have a nice day.

Use the given information to find the exact value of each of the following: $cosine 2 theta$
$\sin θ = \frac{12}{13},$

Key Concepts

Pythagorean Identity

Sign of Trigonometric Functions in Quadrants

Double-Angle Formula for Cosine

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