In Exercises 35–38, find the exact value of the following unde...

Question

In Exercises 35–38, find the exact value of the following under the given conditions:

c. tan(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression, the tangent of A plus B under the given circumstances. Those circumstances are that the sign of A is seven 25th and A is located in the first quadrant and the sign of B is 9 40 firsts and B is located in the second quadrant. We have four answer choices. The first thing I want to recall is the formula that goes with the tangent of A plus B. And that formula is that the tangent of A plus the tangent of B divided by one minus the tangent of A multiplied by the tangent of B is equal to the tangent of A plus B. So, unfortunately, for us, we are not given the tangent of A or the tangent of B off the bat. So we'll have to find it. So we're gonna start with A. So A again, we know is in quadrant one. So everything's gonna be positive. So that's a good thing. What we're given here is that the sign of A is seven 25th recall that that is the ratio of A Y to R and that the tangent of A will be the ratio of Y divided by X. Since we don't know X at this point, we can use the values of Y and R in the Pythagorean theorem to find the value of X. So using X squared plus Y squared equals R squared, we can say X squared plus Y squared. So seven squared equals R squared or squared. So X squared plus 49 equals 625 subtract from both sides. And we get the X squared equals 576. And then we take the square root of both sides and X will equal 24. And we know it's positive because we're in the first quadrant. So now that we have an X value and the Y value, we can say that the tangent of A is Y so seven divided by X which is 24. So the tangent of A is seven 24th. Next, we need to find B. So B again is in the second quadrant. So this means our X value will be negative. We again are given the signs, we have the sign of B is nine 41st. So this is gonna be my Y value divided by my R value. And again, the tangent of B will be the ratio of Y divided by X. So let's find X. So we'll have X squared plus Y squared. So nine squared equals R squared to 141 squared. So X squared plus 81 equals 1681. Subtracting 81 from both sides, X squared equals 1600 taking the square root of both sides X equals 40. But checking back to we're in quadrant two. So X is negative so X is gonna equal negative 40. But now we have the Y value as nine, the X value is negative 40. So we now have a value for the tangent of B. It's negative 9/40. So now let's go back to our formula for the tangent of A plus B and we can substitute things in. So the tangent of A is seven 24th plus the tangent of B which is negative 9/ divided by one minus the tangent of A. So seven 24th multiplied by the tangent of B negative 9/40. We want to create a common denominator for our numerators and it looks like both 24 40 will go into 120 evenly. So that will be my common denominator 24 times five gives me 120. So I'm gonna multiply seven times five and get 40 times three gives me 120. So multiplying negative nine by three, I get negative 27. So I wrote it is minus 27 in the denominator. I'm gonna keep that one put for now minus and then seven multiplied by negative nine is negative 24 multiplied by 40 is 960. So combining what I can in the numerator, 35 minus 27 is eight divided by 120. In my denominator, I'm gonna make a common denominator of and then double negatives are like adding. So when I add 960 I get 1000 and now I want to try to simplify this. I wanna see can I multiply top and bottom by something to get rid of both sets of denominators? And I think I'm gonna multiply both top and bottom. So the full numerator and the full denominator by in my numerator, it works out that 120 goes into 968 times. So we'll just be left with eight, multiplied by eight, which is 64 in the denominator. 960 will cancel with 960 leaving us with 1023. And this looks like it is our exact answer. That 64 divided by 1023 is the exact value of tangent of A plus B in this case. And that's answer choice B have a nice day.

In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

Key Concepts

Sum of Angles Formula for Tangent

Using Sine to Find Tangent

Quadrant and Sign Determination

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