Question

How do you express `sin(pi/12) * cos(pi/4 )` without products of trigonometric functions?

Answer 1

`(sqrt2/4)sqrt(2 - sqrt3)`

Explanation:

`P = sin (pi/12).cos (pi/4)`
Trig table --> `cos (pi/4) = sqrt2/2`.
Then P can be expressed as: `P = (sqrt2/2)sin (pi/12)`
We can evaluate `sin (pi/12)`, using the trig identity:
`cos 2a = 1 - 2sin^2 a`
`cos (pi/6) = sqrt3/2 = 1 - 2sin^2 (pi/12)`
`2sin^2 (pi/12) = 1 - sqrt3/2 = (2 - sqrt3)/2`
`sin^2 (pi/12) = (2 - sqrt3)/4`
`sin (pi/12) = +- sqrt(2 - sqrt3)/2`
Take the positive value because `sin pi/12` is positive
Finally,
`P = (sqrt2/4).sqrt(2 - sqrt3)`