Trig2.3

Find the value of $\cos 105^\circ,\sin 75^\circ\,$

Applying the formula,

$\cos 105=\cos (60+45)=\cos 60 \cos 45-\sin 60 \sin 45\,$

$\cos 105=(\frac{1}{2})(\frac{1}{\sqrt{2}})-(\frac{\sqrt{3}}{2})(\frac{1}{\sqrt{2}})\,$

$\cos 105=\frac{1}{2\sqrt{2}}-\frac{\sqrt{3}}{2\sqrt{2}}\,$

Therefore, $\cos 105=\frac{1-\sqrt{3}}{2\sqrt{2}}\,$

Similarly, $\sin 75=\sin (45+30)=\sin 45 \cos 30+\cos 45 \sin 30\,$

$\sin 75=(\frac{1}{\sqrt{2}})(\frac{\sqrt{3}}{2})+(\frac{1}{\sqrt{2}})(\frac{1}{2})\,$

$\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}\,$

Therefore, $\sin 75=\frac{\sqrt{3}+1}{2\sqrt{2}}\,$