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.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

Accepted Answer

Hey, everyone here we are asked to find the quadrant in which X plus Y lies using trigonometric identities. Here we are given sin of X is equal to 20 divided by 101 sin of Y is equal to negative 21 divided by 29 X as in quadrant one and Y as in quadrant three. Here we have four answer choice options, answer a quadrant one, B quadrant two C quadrant three ND quadrant four. So to begin this problem, we need to start by finding cosine of X and cosine of Y since we are already given sine of X and sine of Y. So beginning with cosine of X, we need to first recall the Pythagorean identity which states that sine squared of X plus cosine squared of X is equivalent to one. Now, we just need to substitute in our given value for sine of X. So we have the quantity of 20 divided by 101 squared plus cosine squared of X is equal to one. Now just solving four cosine of X, we can begin by subtracting this quantity of 20 divided by 101 squared from both sides of the equation. And so we have cosine squared of X is equal to one minus the quantity of 20 divided by 101 squared. Continuing to solve for cosine of X, we now need to take the square root of both sides since we see that cosine of X is squared. And in doing this, we have cosine of X is equal to plus or minus the square root of one minus the quantity of 20 divided by 101 squared. And so now we can notice we have this plus or minus before our square root. And so in order to determine which sign we want for our value, we need to refer to our given information which states that X is in quadrant one. Well, this tells us that cosine of X will then only be positive. And so we can just disregard the negative sign. And so now solving the expression underneath the square root, we have cosine of X is equal to the square root of 9801 divided by 10,000, 201. And so now we just simply take the square root of the numerator and the denominator. And we find that cosine of X is then equal to 99 divided by 101. So now that we have found cosine of X, we can move on to finding cosine of Y using the same method. So recalling again the Pythagorean identity. Now with the Y variable, we have sine squared of Y plus cosine squared of Y is equivalent to one. Now substituting in our given value for sine of Y, we have the quantity of negative 21 divided by 29 squared plus cosine squared of Y is equal to one. Now, using the exact same method as we did for cosine of X, we need to solve for cosine of Y. And so doing this, we have cosine of Y is equal to plus or minus the square root of one minus the quantity of negative 21 divided by 29 squared. And so now, once again, we have this plus or minus before our square root. And so referring to the last piece of given information, we see that Y is in quadrant three. Well, this tells us that cosine of Y will only be negative. So here we disregard the positive sign. Now, just solving the expression underneath the square root, we find that cosine of Y so far is equal to negative square root of 400 divided by 841. Now we just take the square root of the numerator and the denominator. And we find that cosine of Y is equal to negative 20 divided by 29. And so now that we have found cosine of X and cosine of Y, we can move on to finding where X plus Y lies. And we do this by first recalling the sum identity for the sine function, which states that sine of X plus Y is equivalent to sine of X multiplied by cosine of Y plus cosine of X multiplied by sine of Y. And now we can notice we have all of our components for the right hand side of this formula already identified. And so we can simply substitute in these values. And so we have sine of X plus Y is equivalent to 20 divided by 101 multiplied by negative 20 divided by 29 plus 99 divided by 101 multiplied by negative 21 divided by 29. And so now we just need to solve this expression by multiplying the terms together and finding their sum. And in doing this, we find that sign of X plus Y is equal to negative 2479 divided by 2929. And so now, one of our last steps here is to repeat this process. However, now we need to recall the sum identity for the cosine function which states that cosine of X plus Y is equivalent to cosine of X multiplied by cosine of Y minus sine of X multiplied by sine of Y. Once again, we just need to substitute in our values which we already have. And therefore we have that cosine of X plus Y is equivalent to 99 divided by 101 multiplied by negative 20 divided by 29 minus 20 divided by 101 multiplied by negative 21 divided by 29. And now just evaluating this expression, we find that cosine of X plus Y is equivalent to negative 1560 divided by 2929. And now to find where X plus Y lies, we simply need to summarize our findings. So we have found that sign of X plus Y is a negative value or less than one. And we all also found that cosine of X plus Y is also less than zero, less than zero, they're both negative. And from these findings here, we can conclude that we are then in the third quadrant. So from this, we are left here with answer choice C where we have found that X plus Y lies in quadrant three. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Use the given information to find the quadrant of s + t. See Example 3.
sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

Key Concepts

Quadrants of the Coordinate Plane

Sine Function and Its Values

Sum of Angles in Trigonometry

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