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

Question

Use the given information to find sin(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 exact value of sine of X plus Y using a trigonometric identity. 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, we have four answer choice options, answer choices A through D each consisting of some fraction that is either positive or negative. So to begin this problem again, we are asked to find the exact value of sine of X plus Y. Well, this expression indicates that we need to recall 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. So now we can notice that we are given sin of X and given sin of Y. However, we are missing cosine of X and cosine of Y. So we cannot yet find the exact value. So we can begin first by finding cosine of X. And for this, we need to recall the Pythagorean identity, which states that sine squared of X plus cosine squared of X is equivalent to one. So now we just substitute in our given value for sin of X. And so we have the quantity of 20 divided by 101 squared plus cosine squared of X is equal to one. Now we begin solving for cosine of X by first subtracting the quantity of 20 divided by 101 squared from both sides. And doing this, we have cosine squared of X is equal to one minus the quantity of 20 divided by 101 squared. Now we notice that cosine of X is squared. So we need to now take the square root of both sides of the equation. And so 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 now we notice that we have this plus or minus in front of the square root. Well, we are told that X is in quadrant one. So this indicates that cosine of X will only be positive. So we can simply disregard the negatives. Therefore, cos sign of X equals the square root of now evaluating that expression underneath the square root, we have the square root of 9801 divided by 10,201. Now just taking the square root of the numerator and the denominator, we find that cosine of X is equal to 99 divided by 101. Now that we have identified cosine of X, we can move on to finding cosine of Y. And so we find it using the same method recalling the Pythagorean identity, which now states that sine squared of Y plus cosine squared of Y is equal to one. So again, we are given sign of Y. So substituting in that value, we have the quantity of negative 21 divided by 29 squared plus cosine of Y squared or cosine squared of why is equal to one again, solving for cosine of why we first subtract negative 21 divided by 29 quantity squared from both sides. In doing this, we have cosine squared of Y is equal to one minus the quantity of negative 21 divided by 29 squared. Since cosine of Y is squared, we take the square root of both sides and 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 now we have this plus or minus once again in front of the square room. So now we recall the last piece of given information where Y is in quadrant three. Well, this indicates that cosine of Y is negative. So we now disregard the positive sign. Therefore, we have cosine of Y is equal to negative square root and evaluating the expression underneath the square root, we have 400 divided by 841. Now taking the square root of the numerator and the denominator, we find that cosine of Y is equivalent to negative 20 divided by 29. So now that we have found cosine of X and cosine of Y, we can find the exact value of sine of X plus Y by utilizing the formula we recalled at the beginning. So we have sin of X plus Y is equal to sin of X, which is given as 20 divided by 101 multiplied by cosine of Y which we found to be negative 20 divided by 29. We then have plus cosine of X which we found as 99 divided by 101. This is multiplied by sine of Y which is given as negative 21 divided by 29. And now just multiplying the terms together and finding their sum. We find I our final value which is negative 2479 divided by 2929. And with this final fraction, we are left here with answer choice. C where again we have negative 2479 divided by 2929. Thanks for watching. I hope you found this video helpful and I'll see you again, next time.

Use the given information to find sin(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

Sine and Cosine Values in Different Quadrants

Pythagorean Identity

Sine Addition Formula

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