In Exercises 69–70, express the exact value of each function as a...

Question

In Exercises 69–70, express the exact value of each function as a single fraction. Do not use a calculator.

If f(θ) = 2 cos θ - cos 2θ, find f(𝜋/6).

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following expression. Without the help of a calculator, we're gonna write our answer as a single fraction. We are given F F of theta equals three cosine theta minus cosine three theta. We are asked to find F of pi divided by three. Our answer choices are a 2/5 B five halves C two D one half. First thinking about function notation, I'm gonna put pi divided by three in for theta everywhere in my expression. So we will have three cosine pi divided by three minus the cosine of three multiplied by pi divided by three. We'll be using a unit circle to find the exact values of our cosines. But first, I want to multiply three and pi divided by three in my second term. So I will have three cosine pi divided by three minus the cosine. And when I do three multiplied by pi divided by three, I'm left with pi. So my second term is now the cosine of pi from there, I will use my unit circle. I keep the value of three and that's gonna be multiplied by the cosine of pi divided by three, which is one half minus the value for cosine of pi which is negative one. So we have three multiplied by one half minus negative one, three, multiplied by one half is three halves minus negative one is the same as saying plus one. And since we want this as a single fraction, we're gonna make a common denominator. So we have three halves plus two halves and this will add together to be five halves. So the value of that expression is five halves, which is answer choice. B have a nice day.

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In Exercises 69–70, express the exact value of each function as a single fraction. Do not use a calculator. If f(θ) = 2 cos θ - cos 2θ, find f(𝜋/6).