Известно, что sin(π/3+t)+sin(π/3-t)=p. Найдите sin(π/3+t)*sin(π/3-t)

Question

Известно, что sin(π/3+t)+sin(π/3-t)=p. Найдите sin(π/3+t)*sin(π/3-t)

Answer 1

Sin (π/3 + t) + sin (π/3 - t) = 2 sin (π/3)·cos t = 2·√3/2·cos t = √3·cos t = p
cos t = p / √3
cos 2t = 2cos²t - 1 = 2·p²/3  - 1
sin (π/3 + t) · sin (π/3 - t) = 1/2 (cos 2t - cos (2π/3)) = 1/2 · cos 2t - 1/2 · (-1/2) = 
= 1/2 cos 2t + 1/4 = 1/2 · (2·p²/3  - 1) + 1/4 = p²/3 - 3/4