SOLUTION: Use appropriate identities to find the exact value of the expression. tan120(degrees)

Question 596365: Use appropriate identities to find the exact value of the expression.
tan120(degrees)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
When you hear or read "exact value" in Trig, you should immediately think: "I'll need special angles for this."

To solve problems like this one, you start by expressing the given angle in terms of one ore more special angles. Some possibilities for this one are:

tan(2*60)
For this we could use the tan(2x) formula
tan(60+60)
For this we could use the tan(A+B) formula
tan(90+30)
For this we could use the tan(A+B) formula. But we would end up with a tan(90) and tan(90) is undefined. So this will not work.
tan(30+30+60)
There is no tan(A+B+C) formula so the 4th one is not easy. But, if we want to work that hard, we could use tan(A+B) twice. (I'll show you at the end.)<\li>
tan(4*30)
There is no tan(4x) formula. But we could use tan(2x) twice on that one.
So either of the first two look good. I'll do them both:
Using on tan(2(60)), making the "x" a 60 we get:

Using tan(A+B) on tan(60+60), making both A and B 60's we get:

I'm only going to start the last two so you can see ways to handle I-don't-have-a-formula-for-that situations.
Using tan(A+B) twice on tan(30+30+60). First I will designate the 30+30 as the "A" and 60 as the "B":

Using tan(A+B) again on tan(30+30), making the A and B both 30's:

If you replace tan(60) with and tan(30) with this will work out to the right answer! (Try it if you don't believe me!)

Using tan(2x) twice on tan(4(30)). The first time the "x" will be "2*30"

Using tan(2x) again on tan(2*30), making the "x" a 30:

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