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:

c. tan (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 tangent of angle A plus angle B. Using the following information, we know that the cosine of A is negative nine divided by A is in quadrant three. We know that the sign of angle B is negative one divided by three and B is in quadrant four. Our answer choices are answer choice. A negative quantity of 1217 minus square foot two all divided by 476. Answer choice B negative quantity of 1620 minus 1681 square foot two, all divided by 476. Answer choice C, the quantity of 12 plus five square at three all divided by 37. And answer choice D the quantity of five square at two minus 39 all divided by 71. All right, from those answer choices. This one's gonna be fun. All right, we're trying to find the tangent of A plus B. We recall from previous lessons that that is going to be found with in the numerator, we take the tangent of A plus the tangent of B that's all divided by, in the denominator one minus the tangent of A multiplied by the tangent of B. And we've got the cosine of A and the sign of B, how do we get to the tangent of A and the tangent of B? Well, we recall from previous lessons that the tangent of an angle, I'll write the tangent of A is equal to in the numerator, the sign of that angle. So the sign of A divided by the cosine of that same angle. So here, I'm writing the cosine of A. So we've got the cosine of A but not the sign, we've got the sign of B but not its cosine. So we just have to find the sign of A and the cosine of B. So let's work on the sign of angle A. We've got the cosine of A which is equal to negative nine divided by 41. And we know that the cosine is going to be X divided by R. So negative nine is equal to X R is equal to 41. How is that helpful? Well, when we're finding the sign of angle A, that's going to be equal to Y divided by R. And we recall from previous lessons that X squared plus Y squared equals R squared. So if we know X and R, we can get Y, let's work on that. We know X squared is going to be negative nine squared plus Y squared. That's what we need to find equals R squared. R is 41. So 41 squared, negative nine squared is 81 plus Y squared equals 41 squared, which is 1681. Subtracting 81 from both sides, we get Y squared equals 1600. Taking the square root of both sides. We get Y equals plus or minus 40. But we're in looking at our given information, we're in quadrant three. What's Y doing in quadrant three? If we draw a quick sketch of a coordinate plane, we have a vertical Y axis. A horizontal X axis. They come together at the origin in the middle quadrant one is up into the right, then continuing counterclockwise, we have quadrant two up into the left quadrant three, down into the left quadrant four, down into the right in quadrant three, Y values are negative. So we are only talking about the value of negative 40 for Y. So the sign of angle A is going to be in the numerator Y is negative 40 as divided by R which is still 41. OK. So now we can take the sign of angle A negative divided by 41 divided by the cosine of angle A negative nine divided by 41. And we know that that is the same as or the tangent of angle A, our R S are going to be canceling out and we are going to get Y divided by X. So why is negative 40 X is negative nine? Those negatives cancel out. So the tangent of angle A is 40 divided by nine. All right, we've got the tangent of an angle A, we'll be able to fill that in. Now, we need the tangent of angle B. So we know the sign of angle B that's equal to negative one third. We know that the sign is Y divided by R. We essentially just need to figure out our X. So if we've got X squared, that's unknown plus our Y which is negative one squared equals our R which is three squared. Well, we'll have X squared plus negative one squared is just one equals three squared, which is nine, subtracting one from both sides. We get X squared equals positive eight taking the square re of both sides while we get X equals plus or minus for the square root of eight. That's the same as the square rut of four multiplied by two and the square root of four is two. So this simplifies to two square root two. But we're talking about X in quadrant four is X positive or negative. Well, in quadrant four, X is positive. So we're only talking about the positive value of two square root two. So again, here we know that the tangent of angle B is equal to Y divided by X, we know that our Y is negative one and our X is two square root two. But we want to rationalize this, we don't want the square root two in the denominator. So we're going to multiply the whole thing by the squared of two divided by the square root of two. So in the numerator, we have the negative square, it two divided by two multiplied by square root of two multiplied by squared of two. This is two multiplied by two. So that is four in the denominator. So there's our tangent of angle B negative square A two divided by four. Now we just need to fill these in. So let's move this to a different part of our screen where we've got a little bit more room to write and still need to see our tangents. OK. So in the numerator, the tangent of angle A is 40 divided by nine and that's being added to the tangent of angle B which is the negative square at two divided by four. That whole thing is divided by, in the denominator, we have one minus the tangent of angle A which is 40 divided by nine multiplied by the tangent of angle B which is the negative square at two divided by four. And so this one, this one's fun. All right. In the numerator, we want a common denominator to add those two fractions that's going to be 36. So we can multiply the 40 divided by nine by 4/4. And we can multiply the negative square root two divided by four by 9/9. So in the numerator, we'll have 160 divided by 36 plus the negative nine square foot two divided by 36 all divided by, we can do that multiplication. So we'll have one minus in the numerator, the negative 40 square foot two divided by in the denominator 36. Now, we can do a couple of things. We can add our numerators in the numerator. So those actually aren't like terms. So it will just look like this fun minus nine square at two all divided by 36. And now that is divided by, in the denominator, we can get a common denominator to add those. So that's gonna be 36 multiplying the one by 36 divided by 36. So that will be 36 divided by then minus the negative will be plus 40 square at two divided by 36. So we'll have, again, not like terms in the numerators. We'll have 36 plus 40 square. It two all divided by 36. Now, because we're dividing fractions, we can multiply by the reciprocal of the second one and we'll get 160 minus nine square. It two divided by 36 multiplied by 36 divided by 36 plus 40 square foot two. The 36 is canceled out and we'll have in the numerator, 160 minus nine square foot two divided by 36 plus 40 square at two. And now I got this far and I was so satisfied with my answer, but it doesn't match any of the answer choices. Why not? Because we've got this square root two in the denominator and we need to rationalize it. So fun times. Now, we want to multiply by the complex conjugate. So we're going to take our denominator and we're going to multiply it by minus 40 square foot two. And what we do to the denominator, we do to the numerator as well. So multiplying that by 36 minus 40 square foot two and now some fun foiling. So if we're going to foil 1 minus nine square foot two quantity of that by the quantity of 36 minus square foot two. So foiling out our numerators, the 1 60 multiplied by 36 is going to give us 5760. Hopefully, I find all the right parts of my notes here. Now our outsides 160 multiplied by negative square two will be a negative 6400 square foot two. Now the insides negative nine square it two multiplied by 36 is going to be minus 324 square it two. And then our last negative nine square root two multiplied by negative square root two will be plus 720. All right, there's our numerator and in the denominator, we just have to take our squared and that's going to be three oh no, that one is 200 96. And then we'll take our minus the 40 square two squared, which is going to be and simplifying all of that, combining like terms and our numerators and our denominators, we get in the numerator minus 6724 square foot two divided by the negative 1904. And so we'll be able to factor that negative out from the denominator. We'll put that in front and then in the numerator and in the denominator, all of those terms are divisible by four. So we're factoring a four out of the numerator and then it's can it's reducing the denominator. So our simplified answer is in the numerator one. So we have the negative multiplied by the quantity of minus 1681 square at two. That is all divided by 476. Exactly like you would have figured just looking at that problem. Right. Oh my goodness. OK. Let's bring this up to where the answer choices are. Be a mess for a second. OK. So we look at our answer choices and this actually matches with answer choice. B 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:
c. tan (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

Key Concepts

Trigonometric Ratios and Quadrants

Sum of Angles Formula for Tangent

Finding Missing Trigonometric Values Using Pythagorean Identity

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