Question

Find the value of the following

$\left(i\right)\sin {60}^{\circ }\cos {30}^{\circ }+\cos {60}^{\circ }\sin {30}^{\circ }$

$\left(ii\right)\cos {30}^{\circ }\cos {60}^{\circ }–\sin {30}^{\circ }\sin {60}^{\circ }$

Answer 1

(i) Given $\sin {60}^{\circ }\cos {30}^{\circ }+\cos {60}^{\circ }\sin {30}^{\circ }$
$\sin {60}^{\circ }\cos {30}^{\circ }+\cos {60}^{\circ }\sin {30}^{\circ }$

$=\left(\frac{\sqrt{3}}{2}\right)\times \left(\frac{\sqrt{3}}{2}\right)+\left(\frac{1}{2}\right)\times \left(\frac{1}{2}\right)$

$=\frac{3}{4}+\frac{1}{4}$

$=\frac{3+1}{4}$

$=\frac{4}{4}=1$

(ii) Given $\cos {30}^{\circ }\cos {60}^{\circ }–\sin {30}^{\circ }\sin {60}^{\circ }$

$=\left(\frac{\sqrt{3}}{4}\right)-\left(\frac{\sqrt{3}}{4}\right)$

$=0$