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

𝝅 
sec α = ﹣3 , ------ < α < 𝝅
2

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 expression the cosine of beta divided by two where the C can of beta equals negative four. And that beta lies between pi divided by two. And pi we are given four answer choices. The first thing I would like to do is write down what the half angle formula is for cosine. So the one that I'm going to use is that the cosine of beta divided by two is equal to plus or minus the square root of one plus the cosine of beta divided by two. I chose this one because I have the sea can of beta and we know the C of beta equals negative four. And I recall that the cosine of beta will be the same as one divided by the C can of beta. So I can easily find that the cosine of beta is one divided by negative four. So we're gonna use that as negative 1/4. Now that I know that the cosine of beta is negative 14, I'm gonna plug it into my half angle identity. So plus or minus the square root of one plus negative 1/4 divided by two. This will equal plus or minus the square root of 3/4 divided by two. And I'm going to make this into single fractions, not a complex fraction. So I have plus or minus the square root of 3/ divided by two. We're calling my rules of fractions. I'm keeping this under my radical. So we still have plus or minus the square root. But now I have 3/4 multiplied by the reciprocal of two, which is one half. This will give me plus or minus the square root of 3/8. Since we know that beta is in the first quadrant, this will be a positive. So I'm not gonna rate plus or minus anymore. I'm just going to assume this is a positive and I'm splitting my radical to be radical three divided by radical eight. I want to rationalize the denominator. So I no longer have a radical in the denominator. And I do that by multiplying top and bottom by radical eight. And this will give me the square root of 24 divided by eight. I want to see if I can simplify the square root of 24 a little bit further. I know four is a perfect square and does go into that. So if I take the square root of four, I have two multiplied by radical six divided by eight. And when I cross, reduce or reduce the numerator and the denominator, I'm left with radical six divided by four. So the cosine of beta divided by two. When the C can of beta is negative four is the square root of six divided by four. And that's answer choice C have a nice day.

In Exercises 55–58, use the given information to find the exact value of each of the following: α b. cos ------ 2 𝝅 sec α = ﹣3 , ------ < α < 𝝅 2

Key Concepts

Trigonometric Functions

Quadrants and Angle Ranges

Exact Values of Trigonometric Functions

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