Complex exercise

Exercise:

Show that , $\frac{\sin 65-i\cos 65}{-\cos 115+i\sin 115}=-i$ where $i\in \mathbb{C},i=\sqrt{-1}\,$

Solution: Let $Z,{{z}_{1}}\,\And \,{{z}_{2}}\in \mathbb{C}$such that

\(Z=\frac{{{z}_{1}}}{{{z}_{2}}}\) and ${{z}_{1}}=\sin 65-i\cos 65$ ,${{z}_{2}}=-\cos 115+i\sin 115$ 

So ${{z}_{2}}=-\cos 115+i\sin 115=\cos \left( 180-115 \right)+i\sin \left( 180-115 \right)$

            $=\cos 65+i\sin 65={{e}^{i\left( 65 \right)}}$

Also ${{z}_{1}}=\sin 65-i\cos 65=-\left( i\cos 65-\sin 65 \right)=-\left( i\cos 65+{{i}^{2}}\sin 65 \right)$

              $=-i\left( \cos 65+i\sin 65 \right)=-i{{e}^{i\left( 65 \right)}}$ Thus $Z=\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{-i{{e}^{i\left( 65 \right)}}}{{{e}^{i\left( 65 \right)}}}=-i$