SOLUTION: if cos(t)= -1/5 and pi < t < 3pi/2, then tan( pi/3 + t) = ????

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Question 626806: if cos(t)= -1/5 and pi < t < 3pi/2, then tan( pi/3 + t) = ????
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
To find we will be using the tan(A+B) formula:

So we will need and tan(t). Since is a special angle we will be able to find its tangent without a calculator. We will use the given information about t and cos(t) to find tan(t).

Since t terminates in the 3rd quadrant. In the 3rd quadrant tan is positive. (Knowing that tan will be positive means we can ignore signs for the rest of finding tan(t).)

Since tan is opposite/adjacent, we will need those numbers to find it. cos is adjacent/hypotenuse. So we can use a 1 for the adjacent side. But how do we find the opposite side? Answer: The Pythagorean Theorem! Let's call the opposite side "x". Then

Solving for x...

(Remember, we don't care about signs so we can forget the negative square root of 24 which, in this case, is actually the number you want to use!)
Simplifying:

So

Since is a special angle we should know that its tan is . Now that we have tan(t) and we can use tan(A+B):

Substituting in the values we have found we get:

First we multiply in the denominator:

The square root in the denominator will simplify:

This may be an acceptable answer. But it does have a square root in the denominator so you may want/need to rationalize it. To rationalize a two-term denominator like this we take advantage of the pattern. To rationalize we multiply it by :

we from the pattern know how the denominators multiply out. For the numerators we must use FOIL:

Simplifying: