Question #139919
Simplify the following expression

Cos ( x+ Pi over 3) - sin ( x + Pi over 6 )
1
Expert's answer
2020-10-26T12:57:16-0400

cos(x+π3)sin(x+π6)cos(x+\frac{\pi}{3}) - sin(x+\frac{\pi}{6})

cos(x+π3)=cosxcosπ3sinxsinπ3cos(x+\frac{\pi}{3})= cosxcos\frac{\pi}{3} - sinxsin\frac{\pi}{3}

sin(x+π6)=sinxcosπ6+cosxsinπ6sin(x+\frac{\pi}{6})= sinxcos\frac{\pi}{6}+cosxsin\frac{\pi}{6}

cos(x+π3)sin(x+π6)=cos(x+\frac{\pi}{3}) - sin(x+\frac{\pi}{6})= cosxcosπ3sinxsinπ3cosxcos\frac{\pi}{3} - sinxsin\frac{\pi}{3} sinxcosπ6cosxsinπ6-sinxcos\frac{\pi}{6}-cosxsin\frac{\pi}{6}

therefore

cosxcosπ3sinxsinπ3sinxcosπ6cosxsinπ6cosxcos\frac{\pi}{3}-sinxsin\frac{\pi}{3}-sinxcos\frac{\pi}{6}-cosxsin\frac{\pi}{6}

this becomes

12cosx12cosx32sinx32sinx\frac{1}{2}cosx-\frac{1}{2}cosx-\frac{\sqrt3}{2}sinx-\frac{\sqrt3}{2}sinx

03sinx=3sinx0-\sqrt3sinx = -\sqrt3 sinx



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