In Exercises 55–58, use the given information to find the exact v...

Question

In Exercises 55–58, use the given information to find the exact value of each of the following:

α
b. cos ------
2

4
tan α = ------ , 180° < α < 270°
3

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression. Using the provided data, we want to find the value of the cosine of beta divided by two where the tangent of beta equals 12 5th and beta lies between 100 80 degrees and 270 degrees. We are given four answer choices. First, I know that using the tangent of beta, I will be able to find the sign of beta or the cosine of beta. So I'm gonna use the half angle identity for the cosine of beta divided by two that states it equals plus or minus the square root of one plus the cosine of beta divided by two. So now I'm gonna use the information they gave me about the tangent of beta equaling 12 5th to find the cosine of beta. I know that tangent is equivalent to the ratio of Y divided by X. And we know that beta is in the third quadrant. So both X and Y will be negative. So X will equal negative five Y equals negative 12. To find the cosine of beta, we need the ratio of X divided by R currently we don't know R but knowing X and Y, we can find R using the Pythagorean theorem. So X squared plus Y squared equals R squared. So this would be negative five squared plus negative 12 squared equals R squared. The 25 plus 144 equals R squared, 169 equals R squared. Take the square root of both sides get R by itself. An R comes out to 13. Now that I know this, I can find the cosine of beta. So that would be negative 5/ knowing the cosine of beta. I'm going back to my half angle formula. So the cosine of beta divided by two, we said is equal to plus or minus the square root of one plus the cosine of beta divided by two. Since beta divided by two will fall into the second quadrant, this will be a negative. And so I'm gonna put it as negative square root of one plus negative 5/ divided by two. Simplifying my numerator, I still have negative radical and then one plus negative thir uh negative 5/13 is 8/13 divided by two. I want to rewrite this without a complex fraction. So it's the negative square root of 8/ divided by the whole number of two. Using my fraction rules, I get negative square root of 8/ multiplied by the reciprocal. So one half, I'm gonna cross reduce the two from one half with the eight from 8/13. And when I do so I get negative square root of 4/ this would be the negative, then I could do. The numerator is the square of four. The denominator is a square to 13. In the numerator, the square root of four is two. So I'll have negative two divided by the square to 13. I don't want to leave a radical in my denominator. So rationalizing that by multiplying top and bottom by the square to 13, I get negative two radical 13 divided by 13. This is as simplified as this can get. So this is the exact value of the cosine of beta divided by two. It's negative two radical 13 divided by which is answer choice C have a nice day.

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In Exercises 55–58, use the given information to find the exact value of each of the following: α b. cos ------ 2 4 tan α = ------ , 180° < α < 270° 3