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

Question

In Exercises 57–64, find the exact value of the following under the given conditions:c. tan (α + β)5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- .6 2 7 2

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression, the tangent of A plus of B under the given circumstances. So the sign of A equals 9/10 and A is between pi divided by two and pi the tangent of B equals 2/9 and B is between pi and three pi divided by two, we are given four answer choices. So to begin with, I want to identify what the tangent of A plus B equals for a formula. And I recall that that is the tangent of A plus the tangent of B divided by one minus the tangent of A multiplied by the tangent of B. So I need to know the tangent of A and the tangent of B. In our problem, we are told that the tangent of B equals 2/9. So that we know, however, we do not know the tangent of A. What we know about the tangent tangent of A is that it would be the sign of A divided by the cosine of A. And we are told that the sign of A is 9/10. So knowing this information, we can find the cosine of A because recall sine is the ratio of Y divided by R and the cosine would be the ratio of X divided by R. So knowing Y and R, we can use a Pythagorean identity X squared plus Y squared equals R squared. And X squared is what we're trying to look for plus Y is nine. So nine squared has to equal R which is squared. So X squared plus 81 equals subtracting 81 from both sides X squared equals 19. So X looks like it'll be the square root of 19. But first, this is for angle A, an angle A is in the second quadrant since angle A is in the second quadrant X is gonna be negative. So we're gonna use negative radical 19. So our value for cosine is negative radical divided by 10. So now using that to find the tangent of A, we have the sign of A. So 9/10 divided by the cosine of A. So negative radical 19 10th, I can multiply both the numerator and denominator by 10 to cancel out my complex fraction denominators. So the tangent of A equals negative nine divided by radical 19. So now we have both of our tangent values and we are going to plug it into our original formula. So we have the tangent of A which is negative nine divided by radical 19 plus the tangent of B which is two nights divided by one minus the tangent of a. So negative nine divided by radical multiplied by the tangent of B which was 2/ in my numerator, I'm gonna create a common denominator which will be nine radical 19. And in doing so my new numerator here is negative 81 plus two radical 19. And the denominator I'm gonna do the multiplication and that will give me one double negatives are like adding. So I'm gonna say plus and then my nines will cancel out. So one plus two divided by radical 19, I wanna rationalize. Nope, I'm not gonna rationalize yet. I'm sorry, I'm gonna make a common denominator in the denominator. And when I do so, I will have negative 81 plus two radical 19 divided by nine radical 19 divided by radical 19 plus two, divided by radical 19. Now to try to get rid of the complex fractions, I'm gonna multiply both the numerator and the denominator of my fraction by nine radical 19. That will cancel out the full denominator in the numerator. So we have negative 81 plus two radical divided by, in the denominator, it'll only cancel out the radical nineteens. So we have nine multiplied by radical plus nine, multiplied by two which is 18, couple more steps. We're almost there. Now, I want to rationalize my denominator since my denominator is a buy no meal, I'm gonna multiply it both the top and the bottom by a conjugate, which has the same terms with the opposite sign in between them. So both top and bottom of this fraction will be multiplied by nine radical 19 minus 18. And when we do so in my numerator, I have to foil and foiling this out will give me negative times radical 19 plus 100 or sorry, 1458 plus I have to do 18 times 19 which is and then minus 36 radical 19. In the denominator, we will get the difference of squares. So I square the first term and so nine squared is multiplied by radical 19 squared which is 19 and that's minus 18 squared which is 324. So now I want to simplify and combine as many like terms as I can. So combining my like terms, I get negative 765 multiplied by radical 19. And that is gonna be plus 1800. And in my denominator 81 multiplied by is 1539 and subtracting 324 from that gives me 1215. And now there must be a way to reduce this. So I'm gonna start with, I know that negative 802,215 are all divisible by five. So I'm gonna start out reducing this by five. And when I do, I get negative 153 radical plus 360 divided by 243. That still looks a little high. I bet you we can reduce it one more time. Um Using my multiplication rules, I think they're all divisible by nine. So I'm gonna divide every single one of those by nine. I get negative 17 radical 19 plus 40 divided by 27. I don't think there's anywhere else I can reduce this. So this looks like my answer. Um It matches answer choice. D if I rearrange my numerator to be minus 17 times radical divided by 27. So we're gonna say this is our answer. It is answer choice. D have a nice day.

In Exercises 57–64, find the exact value of the following under the given conditions:c. tan (α + β)5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- .6 2 7 2

Key Concepts

Trigonometric Identities

Sine and Tangent Functions

Quadrants and Angle Ranges

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