In Exercises 1–6, use the figures to find the exact value of each trigonometric function.

cos 2θ

Video transcript

Hello, everyone. We are asked to determine the exact value of the trigonometric function. Using the provided figure, we are given a right triangle where the legs are 28 and 45. There's an angle beta in the bottom right hand corner across from the measure of the leg 28. And our hypotenuse is 53. We want to find the value of the cosine of two beta. We are given four answer choices. So beginning with the co sign of two beta. So we have an identity that matches this and it says that the cosine of two beta equals the cosine squared of beta minus the sine squared of beta. So I'm gonna use the values in this figure to find out what the cosine of beta is and the sign of beta. And then we'll work from there. So cosine of beta, recalling that the cosine is the adjacent leg divided by the hypotenuse. Here, we would have 45 as the adjacent leg divided by the hypotenuse of 53. And then the sign of beta would be the opposite leg divided by the hypotenuse. So the opposite from beta is 28 divided by the hypotenuse which is 53. So now I'm gonna plug this into the identity. So the cosine of two beta equals cosine squared of beta. So that's the cosine value we found. So 45 53rd squared minus sine squared beta. So the sine value we found so 28 53rd squared. And when we do these squares, we get 2025 divided by 2809 minus 784 divided by 2809. And since they already have a common denominator, we can subtract. And we now know that the cosine of two beta would be 1241 divided by 2809. So this matches with answer choice A have a nice day.