Question

Evaluate the following:
(i) $sin{78}^{\circ }cos{18}^{\circ }-cos{78}^{\circ }sin{18}^{\circ }$
(ii) $cos{47}^{\circ }cos{13}^{\circ }-sin{47}^{\circ }sin{13}^{\circ }$
(iii) $sin{36}^{\circ }cos{9}^{\circ }+cos{36}^{\circ }sin{9}^{\circ }$
(iv) $cos{80}^{\circ }cos{20}^{\circ }+sin{18}^{\circ }sin{20}^{\circ }$

Answer 1

(i) $sin{78}^{\circ }cos{18}^{\circ }-cos{78}^{\circ }sin{18}^{\circ }$
$\left[sin\right(A-B)=sinAcosB-cosAsinB]$
$=sin({78}^{\circ }-{18}^{\circ })$
$=sin{60}^{\circ }$
$=\frac{\sqrt{3}}{2}$

(ii) $cos{47}^{\circ }cos{13}^{\circ }-sin{47}^{\circ }sin{13}^{\circ }$
$\left[cos\right(A+B)=sinAcosB+cosAsinB]$
$=sin({47}^{\circ }+{13}^{\circ })$
$=cos{60}^{\circ }$
$=\frac{1}{2}$

(iii) $sin{36}^{\circ }cos{9}^{\circ }+cos{36}^{\circ }sin{9}^{\circ }$
$\left[sin\right(A+B)=sinAcosB+cosAsinB]$
$=sin({36}^{\circ }+9^{\circ })$
$=sin{45}^{\circ }$
$=\frac{1}{\sqrt{2}}$

(iv) $cos{80}^{\circ }cos{20}^{\circ }+sin{18}^{\circ }sin{20}^{\circ }$
$\left[cos\right(A-B)=cosAcosB+sinAsinB]$
$=sin({80}^{\circ }-{20}^{\circ })$
$=cos{60}^{\circ }$
=$\frac{1}{2}$