How do you express sin(pi/12) * cos(( 7 pi)/12 ) sin(π12)cos(7π12) without products of trigonometric functions?

2 Answers
Jul 1, 2018

-1/2+sqrt(3)/412+34

Explanation:

sin(pi/12)=(sqrt(3)-1)/(2sqrt(2))sin(π12)=3122
cos((7pi)/12)=-(sqrt(3)-1)/(2sqrt(2))cos(7π12)=3122

sin(pi/12) * cos(( 7 pi)/12 )=-((sqrt(3)-1)/(2sqrt(2)))^2sin(π12)cos(7π12)=(3122)2

-((sqrt(3)-1)/(2sqrt(2)))^2=-1/8(sqrt(3)-1)^2=-3/8+sqrt(3)/4-1/8=-1/2+sqrt(3)/4(3122)2=18(31)2=38+3418=12+34

Jul 1, 2018

color(indigo)(=> (sqrt3 - 2) / 4324

Explanation:

sin (pi/12) * cos ((7pi) / 12)sin(π12)cos(7π12)

https://lo.wikipedia.org/wiki/%E0%BB%84%E0%BA%95%E0%BA%A1%E0%BA%B8%E0%BA%A1https://lo.wikipedia.org/wiki/%E0%BB%84%E0%BA%95%E0%BA%A1%E0%BA%B8%E0%BA%A1

=> (1/2) (sin (pi/12 + (7pi)/12) + (sin (pi/12) - ((7pi)/12))(12)(sin(π12+7π12)+(sin(π12)(7π12))

=> (1/2) (sin ((2pi)/3) + sin (-(pi/2))(12)(sin(2π3)+sin((π2))

=> (1/2) (sin (pi/3) - sin (pi/2))(12)(sin(π3)sin(π2))

=> (1/2) (sqrt3 /2 - 1)(12)(321)

color(indigo)(=> (sqrt3 - 2) / 4324