Question

What is $\sin {105}^o+\cos {105}^o$ equal to?

• A. $\sin {50}^o$
• B. $\cos {50}^o$
• C. $\frac{1}{\sqrt{2}}$
• D. $0$

Answer 1

Answer: (C)

$\frac{1}{\sqrt{2}}$

$\sin \left({105}^{\circ }\right)+\cos \left({105}^{\circ }\right)=\sin (60+45)+\cos (60+45)$

$=\sin 50\cos 45+\cos 60\sin 45+\cos 60\cos 45-\sin 60\sin 45$ $\left[\sin \right(A+B)=\sin A\cos B+\cos A\sin B]$ and $\left[\cos \right(A+B)=\cos A\cos B-\sin A\sin B]$.

$=\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}-\frac{\sqrt{3}}{2\sqrt{2}}=\frac{2}{2\sqrt{2}}=\frac{1}{\sqrt{2}}.$