вычислите.
с полным решением!
номер 26.17 а)
ответ должен быть 1
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Ответил mappku
0
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)=\](https://tex.z-dn.net/?f=sin%28frac%7Bpi%7D%7B6%7D%2Bt%29cos%28frac%7Bpi%7D%7B3%7D-t%29%2Bsin%28frac%7B2pi%7D%7B3%7D%2Bt%29sin%28frac%7Bpi%7D%7B3%7D-t%29%3B%5C%0Aleft%28sinfrac%7Bpi%7D%7B6%7Dcos+t%2Bsin+tcosfrac%7Bpi%7D%7B6%7Dright%29cdotleft%28cosfrac%7Bpi%7D%7B3%7Dcos+t%2Bsinfrac%7Bpi%7D%7B3%7Dsin+tright%29%2B%5C%0A%2Bleft%28sinfrac%7B2pi%7D%7B3%7Dcos+t%2Bsin+tcosfrac%7B2pi%7D%7B3%7Dright%29cdotleft%28sinfrac%7Bpi%7D%7B3%7Dcos+t-cosfrac%7Bpi%7D%7B3%7Dsin+tright%29%3D%5C%0A "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](https://tex.z-dn.net/?f=%3Dfrac%7B1%7D%7B4%7Dleft%28cos+t%2Bsqrt%7B3%7Dsin+tright%29left%28cos+t%2Bsqrt%7B3%7Dsin+tright%29%2B%5C%0A%2Bfrac%7B1%7D%7B4%7Dleft%28sqrt%7B3%7Dcos+t-sin+tright%29left%28sqrt%7B3%7Dcos+t-sin+tright%29%3D%5C%0A%3Dfrac%7B1%7D%7B4%7Dleft%7Bleft%28cos+t%2Bsqrt%7B3%7Dsin+tright%29%5E2%2Bleft%28sqrt%7B3%7Dcos+t-sin+tright%29%5E2right%7D%3D%5C%0A%3Dfrac%7B1%7D%7B4%7Dleft%7Bcos%5E2t%2B2sqrt%7B3%7Dcos+tsin+t%2B3sin%5E2t%2B3cos%5E2t-2sqrt%7B3%7Dsin+tcos+t%2Bsin%5E2tright%7D%5C%0A%3Dfrac%7B1%7D%7B4%7D%284cos%5E2t%2B4sin%5E2t%29%3Dfrac%7B1%7D%7B4%7Dcdot4%28cos%5E2t%2Bsin%5E2t%29%3D1 "=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")
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