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: cos 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the expression using the provided data. We're asked to find coast of two beta. We assigned beta is equal to negative three quarters and beta lies in quadrant three. We're given four answer choices. Option A 18, option B negative 18, option C negative three eights and option D three eights. So we're giving this expression that we're looking for goes up to beta. Now we're also given information about sine beta. OK. So can we write cosine of two beta in terms of sine beta? Well, yeah, we can recall your double angle identity and we know that cosine of two beta is equal to one minus two, multiplied by sine squared beta sign beta is negative three quarters. So we can just go ahead and substitute in that information. This is gonna be equal to one minus two multiplied by negative three quarters squared. And now we've used that double angle identity. We've used the information we were given in the problem. All that's left to do is to simplify this expression we have that this is equal to one minus two multiplied by 9 16. Now we have two in the numerator, we have 16 in the denominator. Both are divisible by two. So we're gonna divide the numerator and the denominator by two. So we can write this as one minus 98. You want to find a common denominator in order to do the subtraction simplify. So we want one to be written with the denominator native of eight, we have to multiply the denominator by eight. If we do it to the denominator, we need to do it by to the numerator. So one is equivalent to eight divided by eight. We're subtracting 9/8 from that. OK? And this gives us a final value for co two beta of negative 18. And that is that exact value that we were looking for. If we compare this to our answer choices, we see that this corresponds with answer choice B that's it for this one. Thanks everyone for watching. See you in the next video.

Use the given information to find the exact value of each of the following: cos 2θ
sin θ = ﹣2/3, θ lies in quadrant III.

Key Concepts

Trigonometric Ratios and Quadrants

Double-Angle Identity for Cosine

Using Pythagorean Identity to Find cos θ

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