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 = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2

Accepted Answer

Welcome back everyone in this video, we want to determine the exact value of the expression, the tangent of A plus B where the sign of A is negative 1/7 the core sign of B is negative 1/7 and the angles A and B are between pi and three pi divided by two. For our answer choices A is one divided by four, multiplied by the spirit of three, B is four multiplied by the spirit of three C is negative one divided by four multiplied by the skirt of three and D is undefined. Now, what do we already know? Well, we know that this is an addition identity 10 A plus B and recall that the tangent of A plus B equals the tangent of A plus the tangent of B all divided by one minus the tangent of A multiplied by the tangent of B. So that's what we know. No, we don't know what the tangent of A is. Neither do we know the tangent of B? It's not given in any of the information we have, but we also know that the tangent of A is the ratio of the sign of A to the core sign of A and the same thing goes for the tangent of B. So you know what this tells us. If we are able to figure out what the, the unknown values are, then we could substitute them to solve a ton. And thus put it into our equation from the information given we know that the sign of A is negative 1/7 and the cosine of B is negative 1/7. So we can use that to solve for the cosine of A and the sign of B to substitute. So let's do that. Let's think, is there anything you know that relates the sign of an angle to its core sign? Well, we know the Pythagorean identity. So I'm gonna come down here a little bit. It's gonna work on both sides and let's use that idea. So recall that the Pt Pythagorean identity tells us that the square of sine A plus the square of cosine A equals one, we can substitute the value of sine A to solve. So this means that the square of sine A which would be negative 1/ squared plus the square of cosine A equals one. Therefore, the square of cosine A. Well, before let me don't let me jump ahead of myself, the square of negative 1/7 would be one 49th. OK. And that means the square of cosine A is equal to one minus one 49th, which is 48 49th. Therefore, if the square of the cosine of is 48 49th, that means the cosine of A is equal to the positive or negative square root of 49. No, we can simplify this even some more because there's an opportunity to get a whole number in our denominator. So let me come out here and continue writing because I can rewrite the cosign up of A as the spirit of 48 divided by the script of 49 which is the square root of 48 divided by seven. Remember all of these plus or minus, we don't know which one it is yet. And I can rewrite the skirt of 48 as the square of 16 multiplied by the square root of three all divided by seven which the square of 16 is four. So I can write this as plus R minus the square root of sorry plus R minus four multiplied by the square of three all divided by seven. So that would be our value of the coin of A. But all we need to do now is to figure out if that's a positive value or a negative value. Recall that we were told Anger B and Anger A was between pi and three pi divided by two. So let's use our unit circle to help us visualize what that really means. So it's as if we have our circle, OK. And on our circle, our angle is between pi and three pi divided by two. Now we're talking about the cosine of the angle. I recall that for the cosine of an angle on the unit circle, the X value is equal to the cosine of our angle and the Y value is equal to the sign of our angle. So if our point is right here, that means our X value is going to be negative along with our Y value. So that tells us finally that the core sign of A is going to be negative four multiplied by the spirit of three all divided by seven. So now we have the value for the cosine of A. Let's see if we can find the value for the tangent of B. So we can use the Pythagorean identity again because that means the square of the sign of B. Yes, the square of the sign of B plus the square of the core sign of B equals one, the square of the core sign of B is negative + squared. OK? And no, guess what guys? Let's let's be smart about it. Let's work smart. Remember that the core, the square of the cores sign of B is the same thing as the square of the sign of A. So since it's the same formula, we're going to end up with the same answer. In other words, the sign of B when we work it down, when we do all of this, like what we did on the left hand side, the sign of B will eventually work down to plus or minus four, multiplied by the skirt of three all divided by seven. Again, we need to figure out if this is positive or negative. Remember that as we said, why? And on our unit circle, why is the sign of the angle? So here it's also between the limits of pi and three pi divided by two. So Sine B will also be negative. Therefore, Sine B equals negative four multiplied by the square of three all divided by seven. So both of those are our answers for the sine of B and the cosine of A. So now we're finally getting there. So that means we can, so we can use it to solve for the tangent of A and the tangent of B. So let's do that. So this tells us, I wonder what color I should write it in. Let me put it in blue. So this tells us that oh tangent of A as in purple, the tangent of A would be equal to the sign of A which is negative 1/7 divided by the core sign of A, which as you can see here, that's negative four multiplied by the skirt of three all divided by seven. No, let's work that down. So that would be well negative multiplied by negative is positive. So that would be 1/7 multiplied by seven all divided by four multiplied by the spirit of three factor out the sevens. So that's one divided by four, multiplied by the square of three. That's the tangent of A. Next, let's find the tangent of B. So the tangent is equal to the sine, the value of sine which is negative four multiplied by the square of three, all divided by seven divided by the core sign of B which we know is negative 1/7. So let's go ahead and multiply try to rewrite this as being multiplied by seven, seven K seven. So the tangent of B is negative four multiplied by the square root of three. So now we found the values for the tangent of A. Let me take that and the tangent of B. So what's left? No, we just need to substitute it into our formula for the tangent of A plus B. So all of this tells us that the tangent of A plus B I which is equal to the tangent of A, that's one divided by four multiplied by the square of three plus the tangent of B which is negative four, multiplied by the square root of three. Oops, sorry, that should be positive guys because remember it was a negative multiplayer ban negative here. How did I miss that? So that's actually or multiplied by the skirt of three. And all of that is divided by one minus the tangent of A which is one multiplied by four divided by the square of divided one divided by four multiplied by the square of three times the tangent of B which is four multiplied by the square of three. Wow, this is a mouthful, right. No, let's go ahead first, let's work out what's going on in the denominator. Noticed that one divided by the, divided by four, multiplied by the square of three is the reciprocal of four multiplied by the square of three. So those will factor out and that is going to give us gonna rewrite our, the same denominate, the same numerator. Sorry. And that's gonna be all over, all divided by one minus one. Well, one minus one is zero. This would work out to be zero. But what do we know about a denominator of zero? This means that our function is undefined because we can't divide by zero. So that would be our final answer. And when we go back to our answer choices undefined is answer choice. D. Thanks for watching guys. I hope this video helped.

In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2

Key Concepts

Sum of Angles Formula for Tangent

Determining Quadrants and Sign of Trigonometric Functions

Using Pythagorean Identity to Find Missing Trigonometric Values

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