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

Question

Use the given information to find sin(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 exact value of sine of X plus Y by using the trigonometric identity. Before we jump into this problem, let's go ahead and quickly recall what the Python identity for sin of X plus Y is sign of X plus Y is given to us as sign of X multiplied by cosine of Y plus cosine of X multiplied by sine of Y. There are four different components we need, in order to find the value of sin of X plus Y, we need the sine values with respect to the X angle and Y angle. And we need the cosine values with respect to the Y angle and X angle. The problem tells us that cosine of X is equal to negative 112 divided by 113. We are also told that sine of Y is equal to 84 divided by 85. So currently, we have the two values cosine of X and sine of Y, but we are missing the values of sine of X and cosine of Y. So what we'll need to do is we'll need to use a Pythagorean identity to solve for the two missing values. Let's go ahead and start by solving for sin of X. Now, the Pythagorean identity we'll be using to solve for sin of X is sine squared of X plus cosine squared of X is equal to one before we plug in any values. Let's solve for sign in this given equation. If we solve for sine, we can start by subtracting cosine squared X to both sides of the equation. This will leave us with sine squared of X equal to one minus cosine squared of X. If we take the square root of both sides of the equation, we are left with sine of X is equal to the positive and negative square root because square roots produce two values of one minus cosine squared of X. Now, since the square root produces two values a positive and negative value, we need to figure out what value do we want? Do we want the positive or negative version? Well, if we take a look at the problem, we are told that the angle of X exists in quadrant two of the unit circle, sine of X is positive in quadrant two of the unit circle. So we'll be taking the positive version of this square root. Furthermore, since we know that cosine of X is equal to negative 112 divided by 113 we can plug in this value into the square root. So sin of X is equal to the positive square root of one minus negative 112 divided by 113. That quantity squared. After algebraically simplifying, we should be left with a sine X value of positive 15 divided by 113. So we have the value for sin of X. But now we need to solve for sine cosine of Y. So let's go ahead and start that process. Now we'll be reusing the same Pythagorean identity. But this time we'll be using the Pythagorean identity with respect to the Y angle. So the Pythagorean identity will be using to solve for cosine of Y will be sine squared of Y plus cosine squared of Y is equal to one. Now, just like the left hand side, we're going to solve for cosine in this Pythian identity, we can start by subtracting sine squared of 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. Furthermore, if we take the square root of both sides of this equation, we are left with cosine of Y is equal to the positive or negative square root of one minus sine squared of Y. Now again, we need to figure out if we want the positive or negative version of this square root. Looking back at the problem, we are told that the angle Y exists in quadrant one cosine is positive in quadrant one. So we'd be taking the positive version of the square root. Furthermore, we know that the value of sine of Y is equal to positive 84 divided by 85. So if we substitute this value into the square root cosine of Y will equal to the positive square root of one minus 84 divided by 85. That quantity squared. After algebraically simplifying this value, we are left with cosine of Y is equal to positive 13 divided by 85. So now we have the missing pieces we need to solve for sign of X plus Y. Let's go ahead and plug in these values now, so again, sign of X plus Y is equal to sin of X which we solved to be 15 divided by 113 multiplied by cosine of Y which was which we solved to be 13 divided by 85 plus cosine of X which is given to us as negative 112 divided by 113 multiplied by sin of Y which was given to us as positive 84 divided by 85. On the left hand side, 15 divided by 113 multiplied by 13, divided by 85 will give us the value of 39 divided by 1921. And on the right hand side, negative 112 multiplied by 113. I'm sorry, negative 112 divided by 113 multiplied by 84 divided by 85 will leave us with the value of 9408 divided by 9605. And after taking the difference, the final answer will be negative 9213 divided by 9605. This is going to be the value of sine of X plus Y. And with that being said, the answer to this problem is going to be B 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 = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Key Concepts

Trigonometric Identities

Quadrants and Signs of Trigonometric Functions

Finding Missing Trigonometric Values

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