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

Question

Use the given information to find tan(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 tangent of X plus Y using a trigonometric identity. Here we are given sine of X is equal to 20 divided by 101 sine 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 choices A through D, each of which consists of some fraction that is either positive or negative. So again, we are asked here to find the exact value of tangent of X plus Y. However, before we can find this value, we first need to find the values for cosine of X and cosine of Y since we are given sin of X and sine of Y. So to do this, we need to recall the Pythagorean identity which states that sine squared of X plus cosine squared of X is equal to one. And now we can just simply substitute in our given value for sin of X and then we can solve for cosine of X. So substituting we have the quantity of 20 divided by 101 squared plus cosine squared of X is equal to one. And now we just need to subtract the 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. And now continuing to solve for cosine of X, we now need to take the square root of both sides since we have cosine squared of X. And so 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. So now we can notice we have this plus or minus before our square root. So we need to refer back to one of our pieces of given information which states that X is in quadrant one. This piece of information tells us that cosine of X will then be positive only. So we can simply disregard the negative sign. And so now just solving the expression underneath the root, we have cosine of X is equal to the square root of 9801 divided by 10,201. Now we just need to take the square root of the numerator and the denominator. And we find that our final answer or value for cosine of X comes out to 99 divided by 101. So now that we have found cosine of X, we can move on to finding cosine of Y. And so we just repeat this process. However, now we recall the Pythagorean identity for cosine of Y. So for the variable Y, we have sine squared of Y plus cosine squared of Y is equivalent to one. Now we just have to substitute in our given value for sine of Y. And so our expression becomes the quantity of negative 21 divided by 29 squared plus cosine squared of Y is equal to one. And now repeating these steps, we took to solve for cosine of X. We find that cosine of Y is equal to plus or minus the square root of one minus the quantity of negative 21 divided by 29 squared. Now, once again, we are faced with this plus or minus before the square root. So we just refer to our last piece of given information which states that Y is in quadrant three. This tells us that cosine of Y will only be negative. So in this case, we will disregard the positive sign in front of square root. And so continuing to solve our expression. Now, in solving the expression underneath the root, we have cosine of Y is equal to negative square root of 400 divided by 841. Now just taking the square root of the numerator and the denominator, we find our value for cosine of Y is negative 20 divided by 29. And now, since we have found our values for cosine of X and cosine of Y, we can move on to finding tangent of X and tangent of Y. So beginning first with tangent of X to find this value, we first need to recall that tangent is equivalent to sine divided by cosine. Therefore, tangent of X is equivalent to sine of X divided by cosine of X. And now we already have our sine of X and cosine of X values. So we just need to substitute them into this expression. And so beginning with sin of X, we have again 20 divided by 101. This is then divided by cosine of X which is 99 divided by 101. And now just evaluating this expression, we find that tangent of X is equivalent to 20 divided by 99. And with this, we have found tangent of X. So we can move on to finding our final component which is tangent of Y. And so once again, we recall tangent of Y is equivalent to sine of Y divided by cosine of Y. Now substituting in our given and found information we have S of Y as negative 21 divided by 29. This is divided by cosine of Y which is negative 20 divided by 29. Now simply evaluating this expression, we find that tangent of Y is equivalent to 21 divided by 20. And now that we have found a tangent of X and tangent of Y, we can move on to finding the exact value of tangent of X plus Y. And so to find this exact value, we need to recall the sum identity for the tangent function since we have the sum of two terms within our tangent expression here. And so recalling the sum identity, we know that tangent of X plus Y is equivalent to tangent of X plus tangent of Y all divided by one minus tangent of X multiplied by tangent of Y. And so now we already have all of our components for this formula. So we can simply substitute in our tangent of X and tangent of Y values into the right hand side of this formula. So we have 20 divided by 99 plus 21 divided by 20 all divided by one minus 20 divided by 99 multiplied by 21 divided by 20. And now we just need to evaluate this expression. And in doing so, we find that tangent of X plus Y is equivalent to 2479 divided by 1560. And with this final simplified answer, we are left here with answer choice. A where again, our final answer is 2479 divided by 1560. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Use the given information to find tan(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

Trigonometric Ratios and Quadrants

Pythagorean Identity

Tangent Addition Formula

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