Use the given information to find sin(s + t). See Ex...

Question

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

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

Accepted Answer

Hello, today we are going to be finding the exact value of sin of X plus Y by using the trigonometric identity. Before we jump into the problem, let's quickly write down the identity for sign of X plus Y sign of X plus Y is given to us as sin of X multiplied by cosine of Y plus cosine of X multiplied by sine of Y. So in order to find the value of sin of X plus Y, we can our sin of X and sin of Y values and our cosine of X cosine of Y values. The problem tells us that cosine of X is equal to negative seven divided by 25 sine of Y is equal to four divided by 25. And both the angles X and Y lie within quadrant two. Currently, we have our cosine of X and our sine of Y value. But that means we need to solve for our cosine of Y and sine of X value. Let's go ahead and start this process. We will start by finding our sign of X value. In order to find the sign of X value, 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 sulfur sign in the given identity. In order to solve for sine, we will be subtracting cosine squared of X to both sides of the equation that will leave us with sine squared of X is equal to one minus cosine squared of X. If we take the square root of both sides of the equation, that will allow us to rewrite the equation as sine of X is equal to the positive and negative square root of one minus cosine squared of X. Now, before we continue, we need to determine whether we want the positive or negative value of our square root value. Well, keep in mind that we are working with the X angle and we are told that the X and Y angle both exist within quadrant two of the unit circle sine is positive in the second quadrant of the unit circle. So we will be keeping the positive value of this square root. Furthermore, we do know that cosine of X is equal to negative seven divided by 25. So using this information, we can rewrite sin of X to equal to the positive square root of one minus the quantity negative seven divided by 25 squared. Algebraically simplifying the inside of the square root will leave us with sin of X is equal to the square root of 576 divided by 625. And further simplifying the square root will leave us with 24 divided by 25. This is going to be the value for sin of X. Now, we will repeat this process in order to solve for cosine of Y. Now just like sin of X, we will be using the same Pythagorean identity. But this time with respect to the Y angle. So the identity we will be using is sine squared of Y plus cosine squared of Y is equal to one just like before we are going to solve for cosine of Y by subtracting sine squared of Y to both sides of the equation that will leave us with cosine squared of Y equal to one minus sine squared of Y. By taking the square root of both sides of this equation, we can rewrite cosine of Y to equal to the positive and negative square root of one minus sine squared of Y. Now just like sign, we need to determine if we want the positive or negative value of the square root. The problem tells us that the Y angle is within quadrant two of the unit circle and cosine is negative in quadrant two of the unit circle. So we will be keeping the negative sign in front of the square root. Furthermore, we know that sin of Y is equal to four divided by 25 using this information, we can rewrite cosine of Y to equal to the negative square root of one minus the quantity four divided by 25 squared by algebraically simplifying the inside of the square root, we are left with cosine of Y is equal to the negative square root of 609 divided by 625. And simplifying the inside of the square root will leave us with the negative square root of 609 divided by 25. This will be the value of cosine of Y now that we have the missing pieces of information we need, we can now solve for sign of X plus Y. So sine of X plus Y is equal to sin of X which is given to us or which we salt for to be 24 divided by 25 multiplied by cosine of Y which we solve to be negative square root 609 divided by 25. We will be adding cosine of X which was given to us as negative seven divided by 25 multiplied by sign of why? Which is given to us as four divided by 25. Multiplying the first two fractions together will leave us with negative 24 multiplied by the square root of 609 divided by 625. And multiplying the right two fractions will leave us with negative 28 divided by 625 we can combine both of these fractions together to give us negative 24 multiplied by the square root of 609 minus 28 divided by 625. And this is going to be the final value for sin of X plus Y. And with that being said, the answer to this problem is going to be a. 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 sin(s + t). See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II

Key Concepts

Trigonometric Angle Sum Identity

Determining Sine and Cosine Values in Quadrants

Using Pythagorean Identity to Find Missing Values

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