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:

a. sin(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 provided data, we want to find the value of the sign of beta divided by two where the CCA of beta equals negative four and beta lies between pi divided by two. And pi our answer choices are a radical 10 divided by four B, radical 10 divided by two C radical six, divided by four D radical six divided by two. So we're actually given quite a bit of information from this data. First, we know where beta lies. So we can say that beta divided by two is gonna be in quadrant one. And so it will be positive next. If we are looking for the sign of beta divided by two, we have a formula that says that it is equal to plus or minus. But we know we only need positive. So I'm just gonna use the positive of the square root of one minus the cosine of beta divided by two. Well, we need to find the cosine of beta. Lucky for us, the cosine of beta is one divided by the sea can of beta. And we were told that the C can of beta equals negative four. So we can plug this in and we can say that the cosine of beta is one divided by negative four or negative 1/4. Now we have enough information to go back and try to find the exact value of this expression. So the sign of beta divided by two is gonna equal the square root of one minus negative 1/ divided by two, creating a common denominator. I'm gonna simplify my numerator and we'll have the square root of five fourths divided by two from there. I'm gonna rewrite this as a division sentence. So I'll have the square root of five fourths divided by two. And I recall that when I divide fractions, I multiply by the reciprocal. So I have the square root of five fourths multiplied by the reciprocal of two, which is one half. And this will give me the square root of 5/8. So we're working through this, we're almost there, but we don't want to leave it here. We're not simplified enough yet. So the sign of beta divided by two is the square of 5/8. Recall that when we have a quotient that we're taking the square root of, we can take the square root of the numerator and denominator separately. So we'll have the squared of five divided by the squared of eight, the square of five cannot be simplified any further. However, the square root of eight can. And I recall that I can break this into factors as long as I keep it under a radical. And I want to factor out a perfect square if I can. So the squared of eight is the same as saying the squared of four multiplied by the squared of two, the square to four is two and the squared of two cannot be simplified. So the squared of eight is the same as two radical two. So we're gonna put that in as our denominator. So we have radical five divided by two radical two, we do not wanna leave square roots in the denominator of a fraction. So to rationalize the denominator, we multiply both numerator and denominator by the squared of two. So that is the radical that is in our denominator. That is what we're gonna multiply by in our numerator. Since they are both square roots, we keep the square root symbol and multiply the numbers underneath to give us the square root of 10. In our denominator, the squared of two multiplied by the squared of two is two and then two multiplied by two is four. So we are left with the sign of beta divided by two equals radical 10 divided by four. And that's as simplified as this can get. So this is our exact value and it 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:
a. sin(α/2)
sec α = ﹣3, 𝝅/2 < α < 𝝅

Key Concepts

Trigonometric Functions and Their Relationships

Unit Circle and Angle Quadrants

Pythagorean Identity

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