Question

How do you find cos(x/2) given sin(x)=1/4?

Answer 1

`cos (x/2) = 0.99`
`cos (x/2) = 0.127`

Explanation:

`sin x = 1/4` . Find cos x
`cos^2 x = 1 - sin^2 x = 1 - 1/16 = 15/16`
`cos x = +- sqrt15/4`
There are 2 values of cos x because if `sin x = 1/4`, x could either be in Quadrant 1 or in Quadrant 2
Use trig identity:
`2cos^2 (x/2) = 1 - cos 2a`
In this case:
`cos^2 (x/2) = 1/2 +- sqrt15/8 = 1/2 +- 0.484`
a. `cos^2 (x/2) = 0.984`
b. `cos^2 (x/2) = 0.016`
a. `cos (x/2) = sqrt(0.984) = 0.99`
b. `cos (x/2) = sqrt(0.016) = 0.127`
Check with calculator.
a. `cos (x/2) = 0.99` --> `x/2 = 7^@27` --> `x = 14^@53`
`sin (14^@52) = 0.25`. Proved
b. `cos (x/2) = 0.127` --> `x/2 = 82^@70` --> `x = 165^@41`
`sin (165^@41) = 0.25`. Proved