求(sin10°)^4+(sin50°)^4+(sin70°)^4的值.

Question

求(sin10°)^4+(sin50°)^4+(sin70°)^4的值.

Answer 1

(sin10°)^4+(sin50°)^4+(sin70°)^4
=(1-cos20°)²/4 +(1-cos100°)²/4 +(1-cos140°)²/4
=3/4-1/2(cos20°+cos100°+cos140°)+1/4(cos²20°+cos²100°+cos²140°)
=3/4-1/2(cos20°+2cos120°cos20°)+1/8(1-cos40°+1-cos200°+1-cos280°)
=3/4-1/2*0 +1/8（3-cos40°+cos20°-cos80°）
=9/8 + 1/8 （cos20°-cos40°-cos80°）
=9/8+1/8(cos20°-2cos60°cos20°）
=9/8