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: $sine 2 theta$

$\cos θ = \frac{24}{25},$

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the exact value of the expression sine of two multiplied by beta where the cosine of beta is 84 85th and beta lies in the fourth quadrant. Now, what do we already know? Well, recall that this is a double anger identity and our double ale identity states that the sign of a double angle is equal to two multiplied by the sign of that angle multiplied by the cosine of that angle. Now, when we look, we notice we already have the cosine of B being 84 85th. So if we can figure out what the sign of beta is, then we'll be able to substitute it and solve. But how are we gonna figure that out? Let's use our unit circle to help us. So I'm going to draw a sketch of the unit circle over here. As a matter of fact, I won't even draw the whole circle. I'm just going to draw the part we care about that is quadrant four. So this will be in the positive X direction and the negative Y direction. And as we said, we already know the cores sign of B is equal to 85th. And the angle beta lies in the quadrant in the fourth quadrant. So let me think it would probably look something like this. Now, think about it, we don't know the angle, but we know that its cosine equals 84 85th require that the cosine of an angle is the ratio of its adjacent side to its hypotenuse. So that tells us since it's 84 85th, the side adjacent to the angle which would be the X value would be 84 and the side that's opposite, that's the, the, the hypotenuse of our that's formed would be 85. So guess what this tells us, this tells us that if we could figure out what this unknown value is for our opposite side, let's call it X, then we will be able to find a sign of beta because the sign of beta is equal to that opposite side X divided by 85. So how are we going to figure that out? Well, this is a right triangle and that means it follows the Pythagorean theorem. So by the Pythagorean theorem, now we're focusing on quadrant four. So you know what I should actually just put my little four here just to remind us what we're working in. So by the Pythagorean theorem, the sum of the squares of the two shorter sides, which would be 84 squared plus X squared is equal to the square of the hypotenuse or the longest side, which would be 85 squared. So now we can solve for X because that means X squared equals 85 squared minus 84 squared. And when you work that out, it will work down to 169. So if X squared is 169 that means X is the square root of 169 which could be positive or negative 13. So which one is it again? We're working in quadrant four. So that means our X values are going to be nega our Y values rather are going to be negative. So this value right here is gonna be negative. So that means our value is negative 13. So since our unknown value is negative 13, this tells us that the sine of beta equals negative divided by 85. Know that we have the sign of beta and the cosine of beta we can solve for the sine of two multiplied by beta because that tells us that the sine of two multiplied by beta is going to be two multiplied by the sine of beta negative divided by 85 multiplied by the cosine of beta which was 84 divided by 85. When we go ahead and multiply, you'll find that our numerator it be negative 2184 and denominator multiplied by 85 is 7225. That means the sign of two multiplied by beta would be answer choice. A negative 2184 divide divided by 7225. Thanks for watching. I hope this video helped.

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

Key Concepts

Trigonometric Ratios and Quadrants

Pythagorean Identity

Double-Angle Formula for Sine

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