In Exercises 35–38, find the exact value of the following unde...

Question

In Exercises 35–38, find the exact value of the following under the given conditions:

d. sin 2α

sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the exact value of the sign of two B where the sign of A is negative 1/7. The core sign of B is negative 1/7 and angles A and B are between pi and three pi divided by two. Our answer choices is A equals negative 49th, B equals 47 49th C equals eight multiplied by the square of three all divided by 49 D equals a negative value of eight multiplied by the square of three all divided by 49. Now, what do we already know? Well, we're trying to figure out the sign of two B and that's a double angle. So recall that our double ale identity states that the sign of two B is equal to two multiplied by sine B multiplied by the core sign of B. We already know the core sine of B is negative 1/7. So if we can figure out the sign of B, we can substitute it into our equation and solve for the sin of two B. So let's see if we can do that. So we need the sign of B can sign A or the cosign of B help us. Well, again, let's go back to what we know. Recall that the Pythagorean identity relates the sign of an angle to its cosine because it says that the square of sine, the square of the sign of an angle plus the square of the cosine of an angle is equal to one. Since we already have the cosine of B, then we can use it to help us to solve for the sign of B. So let's do that. That means the square of the sign of B plus the square of the cosine of B which would be negative 1/7 says negative 1/ squared equals one, negative 1/7 squared is equal to one 49th. So the square of sine B plus one 49th is one, which means that the square of sine B has to be equal to one minus one 49th, which is 48 49th. Therefore, this tells us that the sign of B would be equal to the square root of 48 49. And we know that that result can either be positive or negative. Now, let's take it a step further. Let's simplify some more. Remember that by the laws of indices, we could rewrite the square root of 48 49th as the square root of 48 divided by the square root of 49 the squared of 49 is equal to seven. So it could be plus or minus the square of 48 all divided by seven. And no, we can also rewrite the square root of 48 because we can express the square root of 48. Let me come down here, we can express the square root of 48 as the square root of 16 multiplied by three which that is the square of 16. Again, we apply the laws of indices, the square of 16 multiplied by the square of three all divided by seven. And the spirit of 16 is four. So finally, we can say that the sign of B is equal to a positive or a negative value of four multiplied by the square of three divided by seven. But no, the question is which one is it, is it positive four multiplied by the spirit of three divided by seven or is it negative? Well, to get the answer to that, let's go back to our information given we know that our angle has to be between pi and three pi divided by two. So let's try to figure that thought. So on the unit circle, let's use the unit circle to help us. So on the unit circle, we're saying that we have an angle that is going to be between pi and three pi divided by two. And we're talking about the sign of B for so the sign of the angle B, so that means it's going to be somewhere in this region. Now remember that from the unit circle, recall that Y is equal to the, the Y value is equal to the sign of whatever the angle is. So that means down here, we know that our Y value would be negative. So it would be a negative X value as well and a negative Y value. So you know what that tells us that tells us that the sign of B has to be negative. Therefore, we can take out our positive or negative sign and put negative four multiplied by the squared of three divided by seven. So that would be our answer for a sign of the sign of B. So we've done a lot. No, we know the sign of B. So all we need to do is to substitute it into our expression. So let me come down here, I should have some space left. So therefore that means the sign of two B is going to be equal to two multiplied by the sign of B which is negative four, multiplied by the skirt of three, all divided by seven, multiplied by the core sign of B which we already know is negative 1/7. So let's do some working here. No negative multiplied by negative. That is going to make it positive. So that's going to be two multiplied by four. So that's it multiplied by the spirit of three. And in our denominator, we have seven multiplied by seven. So that is going to be 49. So therefore, the sign of two B is equal to eight multiplied by the spirit of three, all divided by 49 which when we look in our answer choices, that would have been answer choice. C thanks a lot for watching this video and I hope it helped.

In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

Key Concepts

Double-Angle Identity for Sine

Determining the Sign of Trigonometric Functions Based on Quadrants

Using the Pythagorean Identity to Find Missing Trigonometric Values

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