In Exercises 57–64, find the exact value of the following unde...

Question

In Exercises 57–64, find the exact value of the following under the given conditions:

b. sin (α + β)

sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the sign of A plus B using the following information. We know that the sign of angle A is divided by 85 A is in quadrant two. We know that the sign of angle B is 16 divided by 65 B is in quadrant one. Our answer choices are answer choice. A negative 110 divided by 221. Answer choice B 57 divided by 373. Answer choice C negative 21 divided by and answer choice D 117 divided by 373. All right, we recall from previous lessons that to find the sign of angle A plus angle B, we need to find the sign of A multiplied by the cosine of B and add that to the cosine of A multiplied by the sign of B. So we know the sign of A and the sign of B. That's great. We've got a start here but we still need to find the cosine of B and the cosine of A. But we recall from previous lessons that we can get the cosines from the signs. So let's work on that. So we know the sign of angle A is divided by 85. We also know that the sign is equal to Y divided by R. Well, for the cosine, the cosine of angle A here is going to be X divided by R. And we recall from previous lessons that X squared plus Y squared equals R squared. So since we know Y that's 13 and we know R that's 85 we can find X and then plug that in for the cosine. So let's plug in for X squared. We don't know plus Y squared. That's 13 squared equals R squared. That's 85 squared doing that squaring, we have X squared plus 13 squared is equals 85 squared is 7225. Subtracting 169 from both sides, we have X squared equals 7056. Taking the square rut of both sides, we get X equals plus or minus and the square rut of 7056 is 84. But here we are told that A is in quadrant two. What's X doing in quadrant two? If we draw a rough sketch of a coordinate plane, we have a horizontal X axis, a vertical Y axis. They come together at the origin in the middle, labeling a couple of these quadrants up into the right is quadrant one continuing counterclockwise up into the left is quadrant two. What's X doing in quadrant two? Well, it's negative. So we only have to worry about the negative value of 84 here for our X. So for the cosine of angle A X in the numerator that's negative 84 divided by R and the denominator that's 85. All right, we now know our cosine of A. We just need to find the cosine of B. So from the sign of angle B which is 16 divided by 65 that's Y divided by R. For sine, we've got X squared, which we're trying to find plus Y squared is squared equals R squared, which is 65 squared. Doing our squaring X squared plus 16 squared is going to be 256 equals squared, which is 4225. Subtracting 256. From both sides, we have X squared equals 3969. Taking the square root of both sides. We have X equals plus or minus 63 but we know that this is in quadrant one. So what's X doing in quadrant one? It's positive. So we're only talking about the positive value of 63. So for the cosine of angle B, that's X divided by R X is positive 63. That's divided by R which is positive 65. All right, we've got all four parts. Now we can just plug things in. So for the sign of A plus B, we start with the sign of angle A which is 13 divided by 85. And that's being multiplied by the cosine of angle B which is 63 divided by 65. And that product is added to the cosine of angle A which is negative 84 divided by 85 multiplied by the sign of angle B which is 16 divided by 65. OK. Now doing that multiplication. So multiplying the first two fractions 13 divided by by 63 divided by 65 13, multiplied by 63 is going to be divided by 85. Multiplied by 65 is 5525. Adding that two negative 84 multiplied by 16 in the numerators is a negative 1344. And that is likewise divided by 85. Multiplied by 65 is still 5525. We have a common denominator so we can add our numerators. Well, 819 minus 1344 is a negative 525. That's divided by our 5525. Well, everything there is divisible by 25. So this reduces to 21 divided by 221 and it is negative because we've got a negative numerator. So we've got negative 21 divided by 221. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

In Exercises 57–64, find the exact value of the following under the given conditions:
b. sin (α + β)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

Key Concepts

Sine Addition Formula

Determining Cosine from Sine and Quadrant

Quadrant Sign Rules for Trigonometric Functions

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