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

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 sign of beta divided by two. When the C can of beta equals negative four and beta lies between pi divided by two and pi we have four answer choices. The first thing I want to do is look for my half angle identities in terms of sign. So the sign of beta divided by two, I know we have a few options. But knowing that we have ac can I know I can get to the cosine pretty quickly. So I'm gonna use the identity plus or minus the square root of one minus the cosine of beta divided by two. So now I'm gonna use that sea can of beta equaling negative four. And I'm gonna find the cosine of beta because I recall that the cosine of beta is the same as one divided by the sea can of beta. So this means the cosine of beta is one divided by negative four, which we're gonna use as negative 14. Now that I know that value, I can plug it into the formula I found for the half angle of a sign. So also we know that beta is in the first quadrant. So this is gonna be positive. So I'm not gonna bother writing my plus or minus. So I have the square root of one minus the cosine of beta, which is negative four divided by two. So we'll have the square root one divided by negative four would be positive five fourths divided by two. I'm going to rewrite this division without a complex fraction. So I have the square root of five fourths divided by the whole number two. And using my fraction rules, this would be the square root of five fourths multiplied by the reciprocal which is one half. When I multiply those together, we get the square root of 5/8. I'm going to rewrite this with two radicals. So it'd be the square root of five divided by the square root of eight. I want to rationalize my denominator. So I'm gonna multiply both the numerator and denominator by the square root of eight. This gives me a numerator that is the square root of 40. And my denominator is now eight. I want to simplify the square root of 40 if at all possible. And it is possible because I could take the square root of four and get two and that would leave me with the square root of 10 and my denominator remains eight. Now reducing that fraction two and eight will reduce leaving us with the square root of 10 divided by four. So the sign of beta divided by two, when C can of beta is negative four would be the square root of 10 divided by four. And that is answer choice A have a nice day.

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

Key Concepts

Trigonometric Functions

Quadrants and Angle Ranges

Reciprocal Identities

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