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
a. sin ------
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 exact value of sine 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 to begin with. I'm gonna write the half angle identity for the sign of beta over two that I'm gonna be working with. And the one I have chose to work with is that it equals plus or minus the square root of one minus the cosine of beta divided by two. Next, I'm gonna use that the tangent of beta equals 5th to find what the cosine of beta equals. I know tangent is the ratio of Y to X and I know that the beta angle falls in the third quadrant. So that means both X and Y are negative. So X is gonna be negative five Y is gonna be negative 12. And we need to find R to be able to find the cosine of beta because the ratio for cosine of beta is X divided by R So we can use the Pythagorean theorem. So X squared plus Y squared equals R squared and we're gonna find the value of R. So this would be negative five squared plus negative squared equals R squared. So 25 plus 144 equals R squared. 169 would equal R squared. Recall the opposite of a square is a square root. So R is gonna equal 13. And if R equals 13, this means that the cosine of beta equals negative 5/13. Now that we know what the value is for the cosine of beta, we can go back to the formula for the sine of beta divided by two. And we'll start solving for it beta divided by two will fall into the second quadrant. So that means it will be positive. So I'm not gonna keep dragging that plus or minus with me. I'm just gonna work with the positive. So I have the square root of one minus the cosine of beta, which is negative 5/13 divided by two. So this would be the square root of 13th divided by two. I'm going to rewrite this. So it's not a complex fraction. So I have the square root of 18 13th under the radical divided by two using my fraction rules. I know I multiply by a reciprocal. So I have the square root of 18 13th multiplied by one half. I'm gonna cross, reduce the two from one half with the 18. So 18 becomes nine and I will have the square root of 9/13. And when I take the square root of the numerator and the square root of the denominator, I have radical nine divided by radical 13. The square to nine is three divided by the square to 13. I need to rationalize my denominator by multiplying both the numerator and denominator by the square to 13. And this will give me three radical 13 divided by three. And that is our answer. Three radical 13 divided by three which is answer choice a 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 a. sin ------ 2 4 tan α = ------ , 180° < α < 270° 3