Question

вычислите.
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номер 26.17 а)
ответ должен быть 1

Answer 1

26.17 a)
$sin(frac{pi}{6}+t)cos(frac{pi}{3}-t)+sin(frac{2pi}{3}+t)sin(frac{pi}{3}-t);\ left(sinfrac{pi}{6}cos t+sin tcosfrac{pi}{6}right)cdotleft(cosfrac{pi}{3}cos t+sinfrac{pi}{3}sin tright)+\ +left(sinfrac{2pi}{3}cos t+sin tcosfrac{2pi}{3}right)cdotleft(sinfrac{pi}{3}cos t-cosfrac{pi}{3}sin tright)=$
$=frac{1}{4}left(cos t+sqrt{3}sin tright)left(cos t+sqrt{3}sin tright)+\ +frac{1}{4}left(sqrt{3}cos t-sin tright)left(sqrt{3}cos t-sin tright)=\ =frac{1}{4}left{left(cos t+sqrt{3}sin tright)^2+left(sqrt{3}cos t-sin tright)^2right}=\ =frac{1}{4}left{cos^2t+2sqrt{3}cos tsin t+3sin^2t+3cos^2t-2sqrt{3}sin tcos t+sin^2tright}\ =frac{1}{4}(4cos^2t+4sin^2t)=frac{1}{4}cdot4(cos^2t+sin^2t)=1$