# Video: Using the Quotient Identities to Evaluate the Tangent Function of an Angle

Find the value of tan π given (11 cos π β 13 sin π)/(11 cos π + 13 sin π) = 2/3.

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### Video Transcript

Find the value of tan π given 11 cos π minus 13 sin π over 11 cos π plus 13 sin π is equal to two-thirds.

Weβre looking to find the value of tan π given some equation in terms of cosine and sine. So, itβs sensible to simply begin by recalling the relationship between tan, sine, and cosine. We know that tan of π is equal to sin of π over cos of π. So, we need to find a way to manipulate our expression on the left-hand side to be in terms of tan. What weβre going to do is divide everything on the left-hand side by cos of π.

Now, because weβre dividing both the numerator and denominator of the fraction by the same value, weβre essentially creating an equivalent fraction. And that means the fraction on the left-hand side doesnβt actually change in size. 11 cos π divided cos π is just 11. Then, we can say 13 sin π over cos π must be 13 tan π.

Similarly, the denominator of this fraction becomes 11 plus 13 tan π. And of course, this fraction is equal to two-thirds. Now, we just need to solve for tan π. So, weβre going to begin by multiplying everything by three. Thatβs three times 11 minus 13 tan π over 11 plus 13 tan π equals two. Remember, only the numerator of the fraction is multiplied by three since weβre essentially multiplying by three over one.

Next, we multiply everything by the denominator on the left-hand side. Thatβs 11 plus 13 tan π. And our equation is three times 11 minus 13 tan π equals two times 11 plus 13 tan π. Our next step is to distribute each set of parentheses. That gives us 33 minus 39 tan π equals 22 plus 26 tan π.

Letβs next add 39 tan π to both sides of our equation, so that we have 33 equals 22 plus 65 tan π. And then, we subtract 22. And we have 11 equals 65 tan π. Weβre solving for tan π, so our final step is to divide through by 65. And when we do, we find that tan π is 11 over 65.