find tan(Φ/2)?

Question

tanΦ=3
pi<0<((3pi)/2)

the answer is [(square root(10))+1]/ -3, but i don't know how to get the answer

I just need to know why the 3 is negative... THANKS

Answer 1

If you just need to know the reason for the -3:

Since Φ satisfies π < Φ < 3π/2, (Φ/2) satisfies π/2 < Φ/2 < 3π/4. This is in Quadrant II, where x is negative and y is positive. So the tangent of any angle with its terminal side in this quadrant is negative. Why then wasn't the answer written as -[(√10)+1]/3 ? My guess is, the minus sign was put on the -3 to emphasize that for this Φ/2 it's the cosine that's negative.