Question

How do you express `cos(pi/ 3 ) * sin( ( pi) / 8 )` without using products of trigonometric functions?

Answer 1

`P = (1/4)sqrt(2 - sqrt2)`

Explanation:

`P = cos (pi/3).sin (pi/8)`
Trig table --> cos (pi/3) = 1/2.
There for, P can be expressed as `P = (1/2)sin (pi/8)`
We can evaluate sin (pi/8) by using trig identity:
cos 2a = 1 - 2sin^2 a
`cos (2pi)/8 = cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)`
`2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 (pi/8) = (2 - sqrt2)/4`
`sin (pi/8) = +- sqrt(2 - sqrt2)/2`
Since `sin (pi/8)` is positive, then,
`P = (1/4)sqrt(2 - sqrt2)`