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)

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 tangent of A plus B. Using the following information. We know that the sign of angle A is 13 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 11 divided by 220. Answer choice B negative 220 divided by 221. Answer trace C negative 21 divided by and answer choice D 21 divided by 220. All right. So we want to find the exact value of the tangent of A plus B. Well, we recall from previous lessons that the tangent of A plus B is equal to in the numerator, we have the tangent of A plus the tangent of B and that's all divided by one minus the tangent of A multiplied by the tangent of B. So that helps but we only know the sign of angle A and angle B. So how do we get from sine to tangent. Well, we also recall that the tangent of an angle. So I'm just going to write down the tangent of angle A here is equal to the sign of that angle. So when I'm writing sign of angle A divided by the cosine of that angle, so here the cosine of angle A, well, that's great. We know the sign of both of our angles. What about the cosine? Well, we recall from previous lessons, we can figure out the cosign from the sign. So let's work on that. So we do know the sign of angle A, it is here 13 divided by 85. And the sign of any angle is going to be equal to Y divided by R. What we're trying to find the cosine of angle A cosine is going to be equal to X divided by R. We also recall from previous lessons that X squared plus Y squared equals R squared. There we go. We know Y and we know R so we can figure out X. So if we're filling in X squared plus Y squared equals R squared, X squared, we don't know plus Y squared Y is in the numerator that's 13. So 13 squared equals R squared, R is in the denominator that's 85 squared. So let's do that squaring. We have X squared plus 13 squared is 169 equals 85 squared, which is 7225. Subtracting 169 from both sides, we get X squared equals 7056. Take the square re of both sides and we get X equals plus or minus 84. Now, here's where this information is important. We know that angle A is in quadrant two. What's X doing in quadrant two? If we recall the coordinate plane, we draw a rough sketch of it with a vertical Y axis. A horizontal X axis. And quadrant one is right above the X axis up into the right and quadrant two is up and to the left of our origin where they meet in the middle. So quadrant two, what are the X values doing? The X values are negative? So we're only talking about X equaling a negative 84. So only the negative 84 here. So when we talk about the cosine of angle A in the numerator X is negative 84 that's divided by R which is still 85. So now we know the sign and the cosine, we can figure out the tangent going back to the tangent of angle A equals the sign of angle A divided by the cosine of angle A. So the sign here is 13 divided by 85. And instead of writing a gigantic fraction within fractions, I am going to put a division symbol. So 13 divided by 85 is divided by the cosine of angle A which is negative 84 divided by 85 we know that when we are dividing fractions, we can multiply by the second fractions reciprocal. So 13 divided by 85 multiplied by 85 divided by negative 84 these 80 fives cancel out. And then we can put the negative in the numerator. We have negative 13 divided by 84. There's the tangent of angle A, we will be plugging that in to these two spots for our tangent of A plus B formula. Almost getting back to where we need to be. All right, now, we can figure out the tangent of angle B using the sign of angle B. So let's first get the cosine of angle B. The sign of angle B is 16 divided by 65. So 16 is our Y and 65 is our R. We have X squared plus Y squared is 16 squared equals R squared is 65 squared. That'll be X squared plus 16 squared is 256 equals 65 squared, which is 4225. Subtracting 256 from both sides, we get X squared equals 3969. And taking the square root of both sides, we get X equals plus or minus 63. Now we are in quadrant one with angle B quadrant one X is positive. So we're only talking about the positive value of 63. So when we go for our cosine of angle B that's X divided by R X here is positive R is still 65. So now to figure out the tangent of angle B, we're taking the sign which is 16 divided by 65 dividing that by the cosine of angle B which is 63 divided by 65 multiplying by the reciprocal of the second fraction, we'll have 16 divided by 65 multiplied by 65 divided by 63 60 fives cancel out. And the tangent of angle B is 16 divided by 63. All right, we've got the tangent of angle B. Now we can plug in the tangents of angle A and B. Let's bring this down to a different part of the screen where we'll have a little bit more room to write. OK. So filling this in, in the numerator, the tangent of angle A is negative 13 divided by 84. And then that's being added to the tangent of angle B which is 16 divided by 63. That numerator is being divided by one minus the tangent of angle A which is negative 13 divided by 84 multiplied by the tangent of angle B which is 16 divided by 63. Now, there are several ways to go about solving and simplifying this. You could plug all of this straight into a calculator and then you'll have your answer. If you don't have access to a calculator, I'll quickly go through the math to do this. Let's add in our numerator first. We want to have a common denominator between 84 63. That's going to be 252. So we'll multiply negative 13 divided by 84 by three, divided by three. And we'll multiply 16 divided by 63 by four, divided by four. And that will give us three multiplied by negative 13 is a negative 39. That's divided by our denominator of plus 16. Multiplied by four is a positive 64 that's divided by our 252. So there's our numerator and the denominator, let's do the multiplication of the two fractions. So we'll bring over one minus and negative 13 multiplied by 16 is going to be a negative 208. That's divided by 84 multiplied by 63 is 5292. And there are spots to simplify and reduce fractions. Along the way, I'm just going to keep them big and simplify at the end. So now looking at our numerator, we have common denominators within our numerator. So adding negative 39 plus 64 is going to get us to be divided by 252 in our numerator. And then I'm gonna put a division symbol instead of continuing these fractions within fractions, we can do a couple of things in our denominator. We have one minus negative 208 divided by 5292. The minus negative is going to be a plus. So we'll have plus 208 divided by 5292. That one we can get the common denominator of 5292. So it will be 5292 divided by 5292 and adding those num numerators, 5292 plus 208 equals 5000 500. That's still divided by 5292. And so now we will go back to multiplying by the reciprocal of the second fractions. We'll have 25 divided by 252 multiplied by 5292 divided by 5500. All right. Doing that multiplication and then simplification, we get 21 divided by 220 that is the tangent of A plus B. Let's bring this back up to where we can see our answer choices on the screen and we look at our answer choices and this matches with answer choice. D 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)
sin α = 3/5, α lies in quadrant I, and sin β = 5/13, β lies in quadrant II.

Key Concepts

Trigonometric Ratios and Quadrants

Sum of Angles Formula for Tangent

Finding Missing Trigonometric Ratios Using Pythagorean Identity

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