How do you use a half-angle formula to simplify tan 195?

1 Answer
Jun 15, 2015

tan(195^o) = tan(390^o/2)

The identity you need is:

tan(x/2) = pmsqrt((1-cosx)/(1+cosx))
+ if quadrant I or III
- if quadrant II or IV

It's derived from:

sin(x/2) = pmsqrt((1-cosx)/2)
+ if quadrant I or II
- if quadrant III or IV

cos(x/2) = pmsqrt((1+cosx)/2)
+ if quadrant I or IV
- if quadrant II or III

Divide the quadrant conditions to get the sign you need. tan390^o = tan30^o, so you're in quadrant I. Thus, no matter what, the sign of the answer is positive.

tan(390^o/2) = sqrt((1-cos390^o)/(1+cos390^o))

= sqrt((1-cos30^o)/(1+cos30^o))

= sqrt((1-sqrt3/2)/(1+sqrt3/2))

Get common denominators:
= sqrt(((2-sqrt3)/2)/((2+sqrt3)/2))

Cancel:
= sqrt((2-sqrt3)/(2+sqrt3))

Multiply by the conjugate of the denominator:
= sqrt(2-sqrt3)/sqrt(2+sqrt3)*sqrt(2-sqrt3)/sqrt(2-sqrt3)

Simplify:
= (2-sqrt3)/sqrt(2^2-(sqrt3)^2)

= (2-sqrt3)/sqrt(1)

= 2-sqrt3