Question

Without using the trigonometric tables, find
a) $\sin {105}^o$
b) $\cos {105}^o$
c) $\tan {105}^o$

Answer 1

$\sin 105=\sin (90+15)=\cos 15$

$\cos 105=\cos 90+15=-\sin 15$

$\sin 15=\sin (45-30)=\sin 45\cos 30+\sin 30\cos 45$

$\sin 15=\sin (45-30)=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}-\frac{1}{2}\times \frac{1}{\sqrt{2}}$

$\sin 15=\sin (45-30)=0.2588$

$\cos 15=\cos (45-30)=\cos 45\cos 30+\sin 45\sin 30$

$\cos 15=\cos (45-30)=\frac{1}{\sqrt{2}}\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\frac{1}{2}=.965$

$sin105=\sin (90+15)=\cos 15=.965$

$\cos 105=\cos 90+15=-\sin 15=-0.258$

$\tan 105=\frac{\sin 105}{\cos 105}=\frac{0.965}{-0.258}=-3.74$