Question

noob

sin² 20度+sin² 40度+sin 20度×sin 40度=?

Answer 1

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qn

設 s = sin20° , c = cos20°

sin40° = sin(60°-20°)
= sin60° cos20° - cos60° sin20°
= (√3/2) cos20° - (1/2) sin20°
= (√3/2) c - (1/2) s

sin²40° = (3/4) c² + (1/4) s² - (√3/2) s c

sin20° sin40° = (√3/2) s c - (1/2) s²

sin²20° + sin²40° + sin20° sin40°
= s² + [(3/4)c² + (1/4)s² - (√3/2)sc] + [(√3/2)sc - (1/2)s²]
= (3/4)s² + (3/4)c²
= 3/4