Use the given information to find the exact value of each of t...

Question

Use the given information to find the exact value of each of the following: $cosine 2 theta$

$\cos θ = \frac{24}{25},$

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the expression using the provided data. The expression we wanna evaluate is cosine of two beta. We're told that cosine of beta is equal to 21 divided by 29. And that beta lies in quadrant four. We're given four answer choices. Option A is 41 divided by 841. Option B is the same as A but negative. Option C is 841 divided by 41. And option D is the same as C but negative. Let's start by writing what we were given. We wanna evaluate cosign of two beta. Now, we're giving information about cosine of beta. So can we write cosine two beta in terms of cosine beta? It turns out we can't recall our double angle formula. We can write that cosine of two beta is equal to two cosine squared of beta minus one. And we know cosine of beta. We just need to substitute that in and simplify, we can write this as two multiplied by divided by 29 squared minus one. Simplify, we get two multiplied by divided by 841 minus one. Simplifying further, we're gonna multiply that two into our brackets and then we want to find a common denominator. So right now we have a denominator of 841 in the first term. So we want the same in the second term. So we're gonna write that minus one as negative divided by 841. So our expression becomes 800 in divided by 841. Mine is 841 divided by 841. OK? We have this common denominator now. So we can subtract in the numerator. We get that this is equal to 41 divided by 841. And that is the exact value of the expression cosine of two beta that we were looking for. If we compare this to our answer choices, we can see that this corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Use the given information to find the exact value of each of the following: $cosine 2 theta$
$\cos θ = \frac{24}{25},$

Key Concepts

Cosine of an Angle in a Specific Quadrant

Double-Angle Identity for Cosine

Using Pythagorean Identity to Find sin θ

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