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

Question

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

b. cos(α/2)

sec α = ﹣3, 𝝅/2 < α < 𝝅

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression. Using the data provided, we want to find the cosine of beta divided by two where the C can of beta equals negative four and beta is between pi divided by two. And pi our answer choices are a, the square root of 10 divided by four B, the squared of 10 divided by two C, the squared of six divided by four D, the squared of six divided by two. So our data actually gives us a lot of information. We know we want to find the cosine of beta divided by two and we know where beta falls. And through that information, we can say that beta divided by two is in quadrant one. So then it will be positive. So now I can say that the formula for the cosine of beta divided by two is the positive of the square root of one plus the cosine of beta divided by two. I don't have to take into account the negative side because I know beta divided by two is positive. I do need to find the cosine of beta. And I recall that cosine is the same as one divided by C can. So our cosine of beta is gonna equal one divided by the CK of beta. We are told that the C can of beta is negative four. So this means that our cosine of beta will be negative 1/4. And now I think we have enough information to start solving this. So the cosine of beta divided by two equals the square root of one plus negative 1/ divided by two under my radical. I'm gonna create a common denominator for my numerator which will be four. And when I combine that I get the square root of 3/ divided by two, and I'm gonna rewrite this as fraction division. So I have 3/4 under a radical divided by the value of two. And I recall that when I divide with fractions, I multiply by the reciprocal of the second fraction. So I have the square root of 3/4 multiplied by the reciprocal of two, which is one half. So at this step, we have the square root of 3/8. We're not done yet. We want to simplify this a little bit more. So we're gonna continue with the cosine of beta divided by two. It's the squared of 3/8. And I recall with square roots of a quantity or quotient, I can do the squared of the numerator and the squared of the denominator. So I have the squared of three divided by the squared of eight squared of three cannot be simplified. However, the square root of eight can. So if I bring that over to the side, I can break this down into factors and say the squared of eight is the same as the squared of four multiplied by the square of two, the square to four is two, two multiplied by radical two is the same as the squared of eight. So we're gonna put that back in. So now we have radical three divided by two radical two and we need to rationalize our denominator. So we're gonna multiply numerator and denominator by radical two to get it out of the denominator. In my numerator. I recall that when I have two values both under radicals, I can multiply the values underneath. So I'll end up with the square root of six as my numerator in my denominator, the squared of two multiplied by the squared of two is two, two, multiplied by two, is four. So this is as simplified as we can get and it is an exact value. So the cosine of beta divided by two is radical six divided by four. And that is 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:
b. cos(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅

Key Concepts

Secant and Cosine Relationship

Angle Interval and Sign of Trigonometric Functions

Half-Angle Formulas

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