Find the value of tan 𝜃 given 11 cos 𝜃 minus 13 sin 𝜃 over 11 cos 𝜃 plus 13 sin 𝜃 is equal to two-thirds.
We’re looking to find the value of tan 𝜃 given some equation in terms of cosine and sine. So, it’s sensible to simply begin by recalling the relationship between tan, sine, and cosine. We know that tan of 𝜃 is equal to sin of 𝜃 over cos of 𝜃. So, we need to find a way to manipulate our expression on the left-hand side to be in terms of tan. What we’re going to do is divide everything on the left-hand side by cos of 𝜃.
Now, because we’re dividing both the numerator and denominator of the fraction by the same value, we’re essentially creating an equivalent fraction. And that means the fraction on the left-hand side doesn’t actually change in size. 11 cos 𝜃 divided cos 𝜃 is just 11. Then, we can say 13 sin 𝜃 over cos 𝜃 must be 13 tan 𝜃.
Similarly, the denominator of this fraction becomes 11 plus 13 tan 𝜃. And of course, this fraction is equal to two-thirds. Now, we just need to solve for tan 𝜃. So, we’re going to begin by multiplying everything by three. That’s three times 11 minus 13 tan 𝜃 over 11 plus 13 tan 𝜃 equals two. Remember, only the numerator of the fraction is multiplied by three since we’re essentially multiplying by three over one.
Next, we multiply everything by the denominator on the left-hand side. That’s 11 plus 13 tan 𝜃. And our equation is three times 11 minus 13 tan 𝜃 equals two times 11 plus 13 tan 𝜃. Our next step is to distribute each set of parentheses. That gives us 33 minus 39 tan 𝜃 equals 22 plus 26 tan 𝜃.
Let’s next add 39 tan 𝜃 to both sides of our equation, so that we have 33 equals 22 plus 65 tan 𝜃. And then, we subtract 22. And we have 11 equals 65 tan 𝜃. We’re solving for tan 𝜃, so our final step is to divide through by 65. And when we do, we find that tan 𝜃 is 11 over 65.