Use the given information to find the quadrant of s + t. ...

Question

Use the given information to find the quadrant of 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 quadrant in which X plus Y lies. We're told to use trigonometric identities. We're given the fact that cosine of X is equal to negative seven divided by 25 and sine of Y is equal to four divided by 25. X and Y are in quadrant two. We're given four answer choices. Option A quadrant one, option B quadrant two, option C quadrant three and option D quadrant four. So let's start by thinking about what we need to do here in order to figure out what quadrant X plus Y lies, we can figure out some of these trig functions of that angle X plus Y. So we figure out cosine of X plus Y and we figure out sine of X plus Y, we can evaluate the s of those two values whether they're positive or negative. And that's gonna tell us what quadrant this angle lies in. So if we think about finding cosine of X plus Y, our trigonometric identity tells us that cosine of X plus Y is equal to cosine of X multiplied by cosine of Y minus sine of X multiplied by a sine of Y. And similarly, for sin of X plus Y, we can write that this is equal to sin of X multiplied by cosine of Y plus sine of Y multiplied by cosine of X. In order to figure out cosine of X plus Y sine of X plus Y to get us to that information about the quadrant, we need cosine of X which we have cosine of why we don't have that yet. OK. So let's start there, let's find this cosine of Y value. And that actually pops up in both equations. And so we're gonna start by finding co sign of what now recall that S squared, why plus cosine squared Y must equal one. And we have that trigonometric identity. We know sine of Y. OK. So we can substitute that in and solve for cosa doing that. We get four divided by 25 squared with cosine squared of Y is equal to one simplify 16 divided by 625 plus cosine squared Y is equal to what we're gonna keep moving down. We're gonna subtract 16 divided by 625 from both sides. We get that cosine squared of Y is equal to. Now, when we subtract, we're gonna end up with 625 in the denominator. So let's take this one and write it in terms of 625 in the denominator as well. So we have this common denominator that we can simply subtract from. This is gonna give us 625 minus 16 divided by 625. And we can simplify this cosine squared Y is going to be equal to 609 divided by 625. So the last step is to take the square root and when we take the square root, we're gonna get both the positive and the negative root. And we have to figure out which one it is. And in trig problems like this, it's really important to remember, we take the square root, we get both of those roots and we have to interpret and infer which one is correct. Now, we have cosine of Y that we're looking for and we've been told that Y is in quadrant two and recall that in quadrant two, cosine is negative. So we're gonna take the negative root. We get cosine of Y is negative the square root of 609 divided by 625. Oops, sorry, just divided by 25 when we take the square root and again, it's negative because we are in quadrant two and cosine is negative there. I'm gonna put a red box around the value we just found. So we don't lose track of it. We're gonna need that in our equation. Now, let's go back to those equations. We wrote out initially cosine of X plus Y and sine of X plus Y, we have cosine of X. We just found cosine of Y. We have sin of Y, but we need sin of X as well. OK. So the step, second step here is going to be doing the exact same thing, but in order to find sign, OK. So now we're finding sine of X and the information we were given about X is that cosine of X is equal to negative seven divided by 25. We're gonna use the exact same identity sine squared of X plus cosine squared of X is equal to one substituting in our values sine squared of X plus negative seven divided by 25 squared is equal to one time square to X plus 49 divided by 625 is equal to what subtracting on both sides, sine squared of X is equal to 625 minus 49 divided by 625. And we've turned that one into 625 divided by 625. That's the exact same value. We've just written it like that to have that common denominator. And so we have that sine squared of X is equal to 576 divided by 625. And the last step step is to take the square root. And in this case, when we take the square root, we're gonna take the positive root because we're, we have sin and in quadrant two, sine is positive. So sin of X is going to be 24 divided by 25 and it's positive because we are in quadrant two. OK? So let's put a red box around that value as well. We now have the value of cosine of Y and sine of X. And we can get back to finding the values of cosine of X plus Y and sine of X plus Y to determine which quadrant that angle lies cosine of X plus Y is going to be equal to cosine of X negative seven divided by 25 multiplied by cosine of Y negative the square root of 609 divided by 25 minus sine of X 24 divided by 25 multiplied by S of Y four divided by 25. We can simplify this. This is gonna be equal to seven multiplied by the square root of 609 minus 96 all divided by 625. And if we work out this value on our calculator, this is a positive value. OK. So cosine of X plus Y is going to be greater than zero. Now moving on to sine, OK. Sine of X plus Y is going to be equal to sin of X 24 divided by 25 multiplied by cosine of Y negative square root of 609 divided by 25 plus sine of Y four divided by 25 multiplied by cosine of X negative seven divided by 25. And when we simplify this, we have two negative terms negative 24 multiplied by the square root of 609 minus 28 all divided by 625. And because both of our terms are negative, this value sign of X plus Y is going to be negative as well. OK. So what we found is that cosine is positive and sine is neg, OK, cosign X plus Y is greater than zero and sine of X plus Y is less than zero, which tells us the X plus Y must be in quadrant four. Yeah, that's the quadrant where both of these things are true. Let's go to our answer choices and compare what we found. And we found that X plus Y lies in quadrant four which corresponds with answer choice. D Thanks everyone for watching. I hope this video helped you in the next one.

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