Use the given information to find tan(s + t). See Example...

Question

Use the given information to find tan(s + t). See Example 3.

cos s = -1/5 and sin t = 3/5, s and t in quadrant II

Accepted Answer

Hey, everyone in this problem, we're asked to find the exact value of a tangent of X plus Y. And we're told to use a trigonometric identity. We're given the fact that cosine of X is equal to negative seven divided by 25 sine of Y is equal to four divided by 25. And X and Y are in quadrant two. We have four answer choices. Option A, the square root of 609 divided by 11. Option B negative the square root of 609 divided by 11. Option C negative four multiplied by 42 plus the square root of 609 all divided by 33. And option D negative four multiplied by 42 minus the square root of 609 all divided by 33. So let's get started. We're gonna start by writing what we're trying to find. So we're trying to find an exact value for a tangent of X plus Y. They were told to use a trigonometric identity. And when we see a trigonometric function like this, where we have two things added together inside of that trigonometric function, we want to think about using our thumb identity scan or thumb formulas. So tangent of X plus Y recall can be written as tangent of X. Yes, tangent of why divided by one minus tangent of X multiplied by tangent of Y. Now, in order to use this trigonometric identity, we need tangent of X, we need tangent of Y and we don't have those things yet. We have some information about cosine of X and some information about sine of Y. So let's start by figuring out what these values are. So we wanna find tangent of X and we wanna find tangent of Y. So starting with tangent effects and we're gonna find tangent of X and we're gonna use the information we're given about X which is cosine of X is equal to negative seven divided by 25. And that were in quadrant two. So we think about cosine, cosine is the adjacent side divided by the hypotenuse. They were in quadrant two, we know cosine is negative and quadrant too and it's negative because the X values are that adjacent side are gonna be negative in the hypotenuse or that R value is always positive. And so we know that our adjacent side is going to be equal to negative seven and the hypotenuse is going to be equal to 25. OK. So when we're talking about tangent recall that tangent is the opposite divided by the adjacent side. OK. We have the adjacent side we need to find the opposite side and recall that the three sides or these three measurements are related through Pythagorean theorem because they form a nice right triangle. So we have that our hypotenuse squared is gonna be equal to the adjacent side squared was the opposite side squared. Substituting in our values 25 squared is equal to negative seven squared was the opposite side squared. Let's go ahead and simplify. We get 625 is equal to 49 plus the opposite side squared. We're gonna subtract 49 on both sides. We get that the opposite side square is equal to 576. And we're gonna take the square root, we're gonna find the opposite side. And because we're in quadrant two, OK? We know that the opposite side is going to be positive. It's the Y measurement that Y value is in quadrant two is positive because we are above the X axis. OK. So we're gonna take the positive route and we get 24. OK? And again, we can think of this as in quadrant two, we know that tangent is negative, OK? We already have a negative adjacent value. The opposite value is going to be positive. OK. So we can write out our tangent value now and we have that tangent of X is equal to negative 24 divided by seven. And we're gonna put a red box around that because we're gonna wanna come back to that value in just a minute. Now we're gonna go ahead and do the exact same thing. But with tangent of why? Now for a tangent of why, what information are we given about? Why? Well, we're told that sine of Y is equal to four divided by 25. So now we have our opposite side. K sine is opposite, divided by hypotenuse. So our opposite side is going to be equal to four and our hypotenuse is going to be equal to 25. OK. Tangent, we just did this opposite, divided by adjacent. So we need that adjacent side. We can relate these to the Pythagorean theorem. Again, hypo squared is equal to adjacent squared most opposite squared. Substituting in our values 25 squared is equal to the adjacent site squared plus four squared. Simplifying 625 is equal to the adjacent side squared plus 60 and subtracting 16 on both sides, our adjacent side squared is going to be equal to 609. Now we're gonna take the square root and we need to figure out if we're taking the positive or negative route. Now, in this case, we are dealing with the adjacent side or the X value of this angle that's in quadrant two and the X value because we're to the left of the Y axis is going to be negative. OK. So that's that adjacent side. We can also think of this in terms of tangent in quadrant two, we know tangent is negative, our opposite side is positive. So the adjacent side must be negative. OK. So we get that the adjacent side is going to be equal to negative the square root of 609. Again, tangent is negative in quadrant two. And so we get that tangent of why is going to be equal to negative four divided by the square root of 609. And we're gonna rationalize the denominator multiply numerator and denominator by the square root of 609. When we get that, this is negative four multiplied by the square root of 609 divided by 609. And again, we're gonna put a red box around that. So we don't lose track of it. OK? So we have our tangent of X value and we have our tangent of Y value. Let's go up and remind ourselves of what we're doing. And what we're doing is we're trying to find the exact value of tangent of X plus Y. We've written down the formula, the identity that allows us to break this sum up. And we need to, we needed to know tangent of XS and tangent of Y on their own to do that. OK? So now we have that we can go back to this formula and substitute in our values. So now we have that tangent of X plus Y is going to be equal to tangent of X which we found to be negative 24 divided by seven. And that's that value we found in bull who else? Negative four multiplied by the square root of 609 divided by 609. That value we found in green. This is all going to be divided by one minus those two values multiply negative 24 divided by seven multiplied by negative four route 609 divided by 609. Now, what we have on the right hand side looks very, very messy. OK. But we're done with the trig portion of this question. We've used our trigonometric identity. We've used the information of a cosine and sine to find the value tangent. And now we're to the point where this is just a matter of simplifying and again, it looks messy. It's not a great expression, but we know how to simplify expressions like these, right? All right. So what are we gonna do? We're gonna start by finding a common denominator. We're gonna do that in the numerator and in the denominator. So in the numerator, we need to multiply that first fraction by 609. So we get negative 24 multiplied by 609. And in the second term, we need to multiply by seven. So we have negative four multiplied by seven, multiplied by the square root of 609. And all of this will be divided by seven multiplied by 609 that common denominator. Now, that's the numerator, we still have our denominator. So then this entire fraction needs to be divided by the denominator in the denominator. We're gonna find a common denominator again. So this one, we need to multiply by both seven and 609. So we get seven multiplied by 609 minus negative 24 multiplied by negative four, route 609 all divided by seven, multiplied by 609. Right now, we can simplify already quite a bit. So we have this seven multiplied by 609 in the denominator of both our numerator and denominator. And so that term is actually going to divide it. OK. So we can divide it that term. And now we're left with just a simple single numerator, single denominator. It makes it a little bit easier to work with and to see what's going on. So we've eliminated those terms. Let's now simplify everything that's left. So negative 24 multiplied by 609 we get negative 14,000 616. Then we have negative four multiplied by seven. That's gonna give us negative 28. So we're subtracting negative 28 or sorry, we're subtracting 28 multiplied by 609. That's the numerator. This is now divided by seven, multiplied by 609 4002. 163. So we have minus negative 24 multiplied by negative four. This is gonna give us minus 96 and then we still have to multiply by that square root of 609. OK. So step by step, we're getting closer, we need to rationalize this denominator. So we are gonna take this, we are gonna multiply the numerator and the denominator and what we're gonna multiply with it's a fault and we're gonna take that denominator. We're gonna switch the sign. So we're gonna multiply by 4263 plus 96 multiplied by the square of 609. And this is gonna make it so that the square root is no longer in the denominator. So again, in the denominator 4263 minus the square minus 96 multiplied by the square root of 609. And then we're multiplying by 4263 plus 96 multiplied by the square root of 609. Now we're going to expand again a little bit of messy expressions. But when we expand in the numerator, we're gonna get a couple of terms that are just numerical values. We can simplify those. We get negative 60 3 million 945,000. OK? So big number there minus 1,522,500 multiplied by the square root. Of 609 in the denominator and the cross terms are going to add to zero. OK. So we're gonna be left with just numbers without square roots in that denominator. OK. So the first term is gonna be 4263 squared. And then we have minus 96 multiplied by the square root of 6600 and nine squared. Let's carry on simplifying. We just lost our square there. So we have a squirt carrying on simplifying. We're gonna leave the numerator as it is. And in the denominator, we are just going to expand and add those values together. And while we do that, we get the, the denominator, he is going to be 12 million 560,000 625. And now we're gonna find a common device. We're gonna simplify these fractions as much as we can. We have huge numbers in the numerator and the denominator. And while we work this out, what we figure out is that we can simplify this down to the following negative 168 minus four multiplied by the square root of 609 all divided by 33. OK. So finding that largest common device uh No, in our answer choices, we have this value of four. That's kind of factored out of everything in those last couple of answer choices. So we wanna do is factor a four out of our answer choice as well so that it's easier to match with. So if we factor this negative four out, what are we left with? Well, 168 divided by four gives us 42. And then we have plus the square root of 609. And all of this is divided by 33. And so this is our final answer. This is the exact value of tangent of X plus Y. There's no more simplifying that we can do here. Let's go back up and compare this to the answer choices we were given and we can see that this corresponds with answer choice. Thanks everyone for watching. I hope this video helped you in the next one.

Use the given information to find tan(s + t). See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II

Key Concepts

Trigonometric Ratios

Quadrants of the Unit Circle

Pythagorean Identity

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