Answer to Question #139919 in Trigonometry for Michael

Question

Simplify the following expression

Cos ( x+ Pi over 3) - sin ( x + Pi over 6 )

Answer 1

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

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

$sin(x+\frac{\pi}{6})= sinxcos\frac{\pi}{6}+cosxsin\frac{\pi}{6}$

$cos(x+\frac{\pi}{3}) - sin(x+\frac{\pi}{6})=$ $cosxcos\frac{\pi}{3} - sinxsin\frac{\pi}{3}$ $-sinxcos\frac{\pi}{6}-cosxsin\frac{\pi}{6}$

therefore

$cosxcos\frac{\pi}{3}-sinxsin\frac{\pi}{3}-sinxcos\frac{\pi}{6}-cosxsin\frac{\pi}{6}$

this becomes

$\frac{1}{2}cosx-\frac{1}{2}cosx-\frac{\sqrt3}{2}sinx-\frac{\sqrt3}{2}sinx$

$0-\sqrt3sinx = -\sqrt3 sinx$