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 = - 15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Accepted Answer

Hello, today we're going to be finding the quadrant in which the value X plus Y lies in. And we'll be using trigonometric identities to help us solve this problem. Before we jump into the problem, let's quickly take a look at a unit circle. Now, since the problem asks us to find the quadrant in which the value X plus Y lies in. What this means is that the value X plus Y represents an angle somewhere within the unit circle. Keep in mind that every angle within the unit circle has a corresponding terminal point and this terminal point can be represented as cosine comma sine. What this means is that cosine represents the X value of the corresponding point and sine represents the Y value of the corresponding point. What this means is that if we want to determine the location of X plus Y, there's two values that we need to solve for, we need to solve for cosine of X plus Y and we'll need to solve for sine of X plus Y. This will give us the signs thus giving us the location of the value X plus Y. So let's go ahead and write down the trigonometric identities for both of these values cosine of X plus Y is given to us as cosine of X multiplied by cosine of Y minus sine of X multiplied by sine of Y. And the identity for sine of X plus Y is given to us as sine of X multiplied by cosine of Y plus cosine of X multiplied by sine of Y. So in order to solve for these two values, we'll need cosine with respect to the X and yy angle and sine with respect to the X and Y angle. The problem gives us the values for cosine of X and sine of Y. But we are missing the values for cosine of Y and sin of X. So in order to solve for cosine of X plus Y and sine of X plus Y, we'll first need to solve for cosine of Y and sine of X. Let's go ahead and start by solving for sin of X. In order to solve for sin of X, we'll be using the Pythagorean identity sine squared of X plus cosine squared of X is equal to one before we plug in any values. Let's go ahead and solve for sign within this identity. In order to solve for sine, we'll subtract both sides of this equation by cosine squared of X that will leave us with sine squared of X is equal to one minus cosine squared of X. In order to solve for sine, we'll take the square root of both sides of this equation. And that will leave us with sine of X is equal to the positive and negative square root of one minus cosine squared of X. Now, since the square root produces a positive and negative value, we need to figure out if we want to keep the positive version or negative version of this value since we are working with the X angle. Keep in mind that the problem tells us that X is located in quadrant two of the unit circle in quadrant two sine is positive. So we'll be keeping the positive version of this square root. Furthermore, we know that the value of cosine of X is given to us as negative 112 divided by 113. So if we plug in this value sin of X is equal to the positive square root of one minus the quantity negative 112 divided by 113 that quantity squared. And once we algebraically simplify sign of X will equal to positive 15 divided by 113. So that is going to be one of the two missing values we need. Now let's go ahead and solve for cosine of Y. Now, just like sign of X, we'll be using the same pyet urine identity. But this time we'll be swapping out the angle X for the angle Y instead. So to solve our cosine of Y, we'll be using sine squared of Y plus cosine squared of Y is equal to one. Now, just like before we're going to solve for cosine first, before plugging in any values. In order to solve for cosine, we subtract sine squared Y to both sides of the equation that will leave us with cosine squared of Y is equal to one minus sine squared of Y. Next, we'll take the square root of both sides of this equation. And that will leave us with cosine of Y is equal to the positive and negative value of one minus sine squared of Y. Now just like sine, we need to figure out if we want to keep the positive or negative version of this cosine value. Since we are working with the angle Y, the problem tells us that the angle Y is located in quadrant one of the unit circle cosine is positive in quadrant one of the unit circle. So we'll be keeping the positive value for cosine. Furthermore, the problem tells us that the value for sin of Y is positive 84 divided by 85. So if we plug in this value cosign of Y is equal to the positive square root of one minus 84 divided by 85 that quantity squared. And after algebraically simplifying cosine of Y will equal to positive 13 divided by 85. Now that we have the missing values we can solve for cosine of X plus Y and sine of X plus Y cosine of X plus Y is equal to cosine of X, which was given to us as negative 112 divided by 113 multiplied by cosine of Y which we solved to be 13, divided by 85 minus sine of X which we solve to be 15, divided by 113 multiplied by sin of Y, which was given to us as 84 divided by 85. After algebraically simplifying this equation, the final value for cosine of X plus Y is given to us as negative 2716 divided by 9605. We'll come back to this value later. Now let's go ahead and solve for sine of X plus Y. Now again, sin of X plus Y is given to us as sin of X which we solved to be 15 divided by 113 multiplied by cosine of Y which we solved to be 13 divided by 85 plus cosine of X which was given to us as negative 112 divided by 113 multiplied by sign of Y which was given to us as 84 divided by 85. After algebraically simplifying this equation, the value of sine of X plus Y is given to us as negative 9213 divided by 9605. So let's go ahead and put these two values into the form of an ordered pair. Now again, any terminal point located on the unit circle can be written as cosine comma sign. Now, since the value of cosine of X plus Y is negative and the value of sine of X plus Y is negative as well. What this means is that the form of the ordered pair is going to be negative comma negative on the unit circle, the quadrant in which both the X and Y values are negative are going to be in quadrant three. And what this means is that the value of X plus Y is going to lie within the third quadrant of the unit circle. And that is going to be the answer to this problem. And with that being said, the overall answer is going to be c so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Use the given information to find the quadrant of s + t. See Example 3.
cos s = - 15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Key Concepts

Trigonometric Ratios and Signs in Quadrants

Sum of Angles and Quadrant Determination

Using Pythagorean Identity to Find Missing Ratios

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