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)

cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

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're told that the cosine of A is negative nine divided by 41. And that A is in quadrant three and the sign of B is a negative one divided by three and that B is in quadrant four. Our answer choices are answer choice A in the numerator, the quantity of five minus seven square at two and that's divided by 123. Answer choice B 53 divided by 246. Answer choice C in the numerator, the quantity of nine minus 80 square at two divided by 123. And answer choice D in the numerator, the quantity of 11 square root two plus five divided by 246. All right. So what do we know? And what can we figure out from what we know? We know from previous lessons that to find the sign of the quantity of A plus B, we know that's equal to the sign of A multiplied by the cosine A B plus the cosine of A multiplied by the sign of B. And we know some of these things, we know the cosine of A and we know the sign of B but we don't know the sign of A or the cosine of B. Well, thankfully, we can figure those out from the information that's given. So let's start on angle A, we know that the cosine of A is negative nine divided by 41. And we know the cosine is X divided by R. We also know from previous lessons that X squared plus Y squared equals R squared. So we can fill some of that in. We know that our X is negative nine, those are both in the numerators. So we take negative nine and square it. And then plus Y squared, we don't know that one yet. That's equal to R squared. R is in the denominator. So that's 41 squared. Well, we have enough information to solve for Y which is great because when we're trying to find the sign of A that is going to be Y divided by R. So we'll be able to find the sign of A, all right, let's solve for Y. So negative nine squared, that's a positive 81. Then plus Y squared equals 41 squared is going to be 16 81. We subtract 81 from both sides and we get Y squared equals 1600 taking the square root of both sides Y equals plus or minus 40. But we have more information. We're told that angle A is in quadrant three are Y values positive or negative in quadrant three. Well, they are negative and quadrant three. So we know that Y here is equal to just the negative 40. So we can put in our sign of A, that's going to be a negative 40 in the numerator for the Y divided by R which is still 41. There's our sign of A. So we've got three of these figured out, we just need to figure out the cosine of B. So very similarly, we know that the sign of B which is given to us is negative one divided by three. And sine, as we said is Y divided by R. So we don't know our X squared, we can put X squared plus our Y squared is negative one squared equals R squared, which is three squared. Simplifying we still don't know our X squared, but negative one squared is positive ones. We have X squared plus one equals and three squared is nine. We can subtract one from both sides and we get X squared equals eight, taking the square rut of both sides. We get X equals plus or minus. And the square root of eight, we know that's the same as the square rut of four multiplied by two, both inside the radical and the square of four is two. So this is two square at two. And we know that we are in quadrant four with this angle is X positive or negative in quadrant four, X is positive. So we're just going to take the positive value here of two square at two. So we go back and we fill out our cosine of angle B, our cosine again, cosine is X divided by R. Now we know our X is positive two square, it two divided by R is three. So now we are going to fill out in our sign of A plus B that equals the sign of A which is negative 40 divided by multiplied by the cosine of B which is two square two divided by three. And then we're adding the cosine of A which is negative nine divided by multiplied by the sign of B which is negative one third. No, we just need to simplify here. So we'll do the first bit of multiplication where we have negative divided by 41 multiplied by two square root two divided by three multiplying in our numerators. We're going to have a negative 80 square root two and that's divided by 41. Multiplied by three is 123. Add that to the second one will do the multiplication of negative nine divided by 41 by negative one divided by three. The numerators negative nine multiplied by negative one is a positive nine divided by 41 multiplied by three is still 123. Now, we can do the addition. So negative 80 square foot two plus nine. Well, those aren't like terms. So we can just put our nine first nine minus 80 square it two. That's over our common denominator of 123. And that is as simplified as this one gets. Now, 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)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

Key Concepts

Trigonometric Angle Sum Identity

Determining Signs of Trigonometric Functions by Quadrant

Using Pythagorean Identity to Find Missing Values

Watch next