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
c. tan ------
2

4
tan α = ------ , 180° < α < 270°
3

Accepted Answer

Mhm Hello everyone. We are asked to determine the exact value of the expression using the provided data. The expression we want to find the value of is the tangent of beta divided by two where the tangent of beta equals 12 5th and beta lies between 80 degrees and 270 degrees. Our answer choices are a 3/4 B negative four thirds C four thirds D negative three halves. The first thing I want to do is write the formula for a half angle for a tangent. So the tangent of beta divided by two will equal plus or minus the square root of one minus the cosine of beta divided by one plus the cosine of beta. So I notice I do not know the cosine of beta. However, what I do know is the tangent of beta and the tangent of beta is 12 5th. And this is like saying it is the ratio of Y divided by X. So I can try to find my hypotenuse using the Pythagorean theorem. So I know X will be five. However, beta lies between 100 80 degrees and 270 degrees. So this is in quadrant three and then quadrate three, X is gonna be negative. So X is negative five also in quadrant three. Why is negative? So Y is gonna equal negative 12, we need to find R which will be our hypotenuse. So using the Pythagorean theorem, so X squared plus Y squared equals R squared, I can say negative five squared plus negative 12 squared equals R squared. So 25 plus 144 equal R squared and this would be equals R squared. And when I take the square root of both sides, R will equal 13. Now that I know this, I can go back and find the cosine of beta because the ratio for the cosine of beta is X divided by R. So this will be negative five divided by 13. And now that I know the cosine of beta, I'm gonna plug that back into my entire formula. So the tangent of beta divided by two equals plus or minus the square root of one minus the cosine of beta divided by one plus the cosine of beta. Since beta divided by two will be in the second quadrant. I know that this is going to be negative. So I'm gonna just keep a negative sign out front as I do the rest of the problem. So plugging in the cosine of beta, we have one minus negative 5/13 divided by one plus negative 5/13. So simplifying this, I have the negative square root of in my numerator, double negatives are like adding. So when I combine this, I will get 18th 13th. And in the denominator, when I subtract, I will get 8/ so I now have the negative square root of 18 13th divided by 8/13. Since they have the same denominator, both the numerator and the denominator, they're complex fractions. I can multiply both the top and the bottom by 13 to simplify these fractions. So I have negative radical 8th and I'm gonna reduce that fraction by two. So I have the negative square root of 9/4. I'm gonna keep the negative out front and then I'm gonna take the square root of nine divided by the square root of four. And this will give me negative three halves. So the exact value of the tangent of beta divided by two where the tangent of beta is 12 5th is negative three halves which is answer choice. D 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 c. tan ------ 2 4 tan α = ------ , 180° < α < 270° 3