Question

How do you express `cos(pi/ 4 ) * sin( ( 15 pi) / 8 )` without using products of trigonometric functions?

Answer 1

`- (sqrt2/2)(sqrt(2 - sqrt2)/2)`

Explanation:

Find `cos (pi/4`) and `sin ((15pi)/8)` separately
Table for trig functions of Special Arcs gives --> cos (pi/4) = sqrt2/2.

`sin ((15pi/8) = sin (-pi/8 + 2pi) = -sin (pi/8).`
Find sin (pi/8)
Use trig identity: `cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)`
`2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin (pi/8) = sqrt(2 - sqrt2)/2` (pi/8 is in Quadrant I)
Finally
`cos (pi/4).sin( (15pi)/8) = (sqrt2/2)(-sin (pi/8)) =*`
`= - ((sqrt2)/2)(sqrt(2 - sqrt2)/2)`