Question

How do you find the exact values of `sin(u/2), cos(u/2), tan(u/2)` using the half angle formulas given `sinu=5/13, pi/2<u<pi`?

Answer 1

`sin u = 5/13`. First, find cos u.
`cos^2 u = 1 - sin^2 u = 1 - 25/169 = 144/169` --> `cos u = +- 12/13`.
Since u is in Quadrant 2, then, cos u < 0.
`cos u = - 12/13`.
To find `cos (u/2)`, apply trig identity:
`2cos^2 (u/2) = 1 + cos u = 1 - 12/13 = 1/13`
`cos^2 (u/2) = 1/26` --> `cos (u/2) = +- 1/sqrt26`.
Sin u is in Quadrant 2, then `u/2` is in Quadrant 1, and `cos (u/2)` is positive:
`cos (u/2) = 1/(sqrt26) = sqrt26/26`
To find `sin (u/2)`, apply the trig identity:
`sin u = 2sin (u/2).cos (u/2)`
`sin (u/2) = (sin u)/(2cos (u/2)) = (5/13)(sqrt26/2) = (5sqrt26)/26`
`tan (u/2) = (sin (u/2))/(cos (u/2)) = ((5sqrt26)/26)(sqrt26) = 5`