How do you express sin(pi/ 4 ) * sin( ( 5 pi) / 3 ) sin(π4)sin(5π3) without using products of trigonometric functions?

1 Answer
Jul 1, 2018

color(violet)(=> (sin (pi/3) - sin (pi/6))/2sin(π3)sin(π6)2

Explanation:

![https://lo.wikipedia.org/wiki/%E0%BB%84%E0%BA%95%E0%BA%A1%E0%BA%B8%E0%BA%A1](useruploads.socratic.orguseruploads.socratic.org)

sin A cos b = (1/2) (sin (A+B) + sin (A-B)), IdentitysinAcosb=(12)(sin(A+B)+sin(AB)),Identity

sin (pi/4) * sin ((5pi)/3)sin(π4)sin(5π3)

=> (1/2) (sin (pi/4 + (5pi)/3) + sin (pi/4 - (5pi)/3))(12)(sin(π4+5π3)+sin(π45π3))

=> (1/2) (sin ((8pi)/12) + sin -(pi/6))(12)(sin(8π12)+sin(π6))

=> (sin ((2pi)/3) - sin (pi/6))/2sin(2π3)sin(π6)2

color(violet)(=> (sin (pi/3) - sin (pi/6))/2sin(π3)sin(π6)2