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:

a. sin(α + β)

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 expression, the sign of A plus B where both the sign of A and the core sign of B equals negative 1/7 and both angles A and B are in the domain pi to three halves of pi for our answer choices A is zero, B is negative one C is one and D is the negative value of four multiplied by the square of three all divided by seven. No, we're trying to figure out the sign of A plus B here and recall that the sign of A plus B equals the sign of A multiplied by the core sign of B plus the sign of B multiplied by the core sign of A. From the information we are given here, we already know the sign of A and we already know the core sign of B no, we just need to figure out what the sign of B equals and the co sign of A equals. So let's think, is there anything that you know that relates the sign and the core sign of an angle? Well, we can also recall the Pythagorean identity, which tells us that the square of the sign of an angle and plus the square of the core sign of that angle equals one. So we can use this idea to solve for the sign of B and for the cosine of A, let's first start by trying to solve for the sign of being. So by using that rule, this tells us that the square of sine B plus the square of cosine B equals one. We don't know what the square of sine B is, but we know that the square, we know that the cosine of B equals negative 1/7. So we can replace that in our formulas as negative 1/ squared equals one negative 1/ squared equals 1 49th. OK? And that means then that the square of Sine B equals one minus 1 49th and one minus 1 49th equals 48 49th. If the square of Sine B equals 48 49th, then I can square root both sides to find the value of sine B. The square cancels the square root on the left hand side that leaves me with sine B. So Sine B equals plus or minus the square root of 49th. We're almost there. We just need to figure out which value of sine B are we going to use the positive value or the negative value? Well, let's use our unit circuit to help us recall that we said B was in the domain between pi and three halves of pie. So on our diagram B would be somewhere in the third quadrant. And what do we know about the sign in the third quadrant? Well, recall that for the unit circle, the X and Y coordinates are in the form the cosine of theta and the sign of theater. And in the third quadrant, the sign of theta would have been negative. Therefore, the sign of B would have to be negative. So that means the value for the sign off B we're going to use is nega the negative value of the square root of 49th. No, we have a value for the sign of B. Let's use the same idea to get a value for the core sign of A, using the same formula that means the square of sine A plus the square of cosine A equals one, we know that the sign of A equals negative 1/7. So we can replace that with negative 17 squared plus the square of cosine A equals one. Now look at both sides, look at what we were doing earlier. Don't you notice that both of them seem similar, both of them have substituted negative 1/7 into our equation, which means that also tells us that the square of the cosine A will also eventually work down to 48 49th. So since the square of cosine A equals 48 49th. It will also be equal to a positive or a negative value of 48 49th. That is the cosine of a. Now, will it be a positive or a negative value? Again, let's use our unit circle on our unit circle using what we know about the coordinates in the third quadrant where A would also have been because A was also between pi and three halves of pi in the third quadrant, the core sign is also negative. Therefore, the value for the cosine of A we're going to use will be negative square root of 48 49th. No, we're almost there now that we have the values of the cosine of it and we have the sign of B, we can substitute them into our expression because that node tells us that the sign of A plus B will be equal to the sign of A which is negative 1/7 multiplied by the core sign of B which is negative 1/7 plus the sign of B which is a negative value of the square root of 49th multiplied by the cosine of A which is a negative value of 48 49th, no negative 1/7 multiplied by negative 1/7 is negative 1/7 squared which is 1 49th. And if we multiply the, the negative square root of 48 49th by itself, then we're going to get a positive value of 48 49th and 1 49th plus 48 49th equals 49th or simply one. Therefore, the sign of A plus B equals one for sign A and the cosine of B equals negative 1/7. When we look back on our answer choices that would have been answer choice. C Thanks a lot for watching everyone. I hope this video helped.

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

Key Concepts

Sum of Angles Formula for Sine

Determining Cosine and Sine from Given Values and Quadrants

Understanding Angle Measures and Quadrants

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