Question

How do you find the exact values of `sin(u/2), cos(u/2), tan(u/2)` using the half angle formulas given `cosu=3/5, 0<u<pi/2`?

Answer 1

`sin (u/2) = sqrt5/3`
`cos (u/2) = (2sqrt5)/5`
`tan (u/2) = 1/2`

Explanation:

Use trig identities:
`2cos^2 (u/2) = 1 + cos 2u`
`2sin^2 (u/2) = 1 - cos 2u`
In this case:
`2cos^2 (u/2) = 1 + 3/5 = 8/5`
`cos^2 (u/2) = 8/10 = 4/5`
`cos (u/2) = +- 2/sqrt5 = +- (2sqrt5)/5`
Since u is in Quadrant I, then `cos (u/2)` is positive.
`2sin^2 (u/2) = 1 - 3/5 = 2/5`
`sin^2 (u/2) = 2/10 = 1/5`
`sin (u/2) = +- 1/sqrt5 = +- sqrt5/5`
Since u is in Quadrant I, then `sin (u/2)` is positive.
`tan (u/2) = sin/(cos) = (sqrt5/5)(5/(2sqrt5)) = 1/2`