A) \[\cos n\left( \frac{\pi }{2}-\theta \right)+i\sin n\left( \frac{\pi }{2}-\theta \right)\]
B) \[\cos n\left( \frac{\pi }{2}-\theta \right)-i\sin n\left( \frac{\pi }{2}-\theta \right)\]
C) \[\cos n\theta +i\sin n\theta \]
D) \[\sin n\theta +i\cos n\theta \]
Correct Answer: A
Solution :
\[{{\left( \frac{1+\sin \theta +i\cos \theta }{1+\sin \theta -i\cos \theta } \right)}^{n}}\] \[={{\left[ \frac{1+\cos \left( \frac{\pi }{2}-\theta \right)+i\sin \left( \frac{\pi }{2}-\theta \right)}{1+\cos \left( \frac{\pi }{2}-\theta \right)-i\sin \left( \frac{\pi }{2}-\theta \right)} \right]}^{n}}\] \[={{\left[ \frac{\begin{align} & 1+2{{\cos }^{2}}\frac{1}{2}\left( \frac{\pi }{2}-\theta \right)-1+i2\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \\ \end{align}}{\begin{align} & 1+2{{\cos }^{2}}\frac{1}{2}\left( \frac{\pi }{2}-\theta \right)-1-i2\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.\cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \\ \end{align}} \right]}^{n}}\] \[={{\left[ \frac{2\cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)\left\{ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)+i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right\}}{2\cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)\left\{ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)-i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right\}} \right]}^{n}}\]\[={{\left\{ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)+i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right\}}^{n}}\] \[{{\left\{ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)-i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right\}}^{-n}}\] \[={{\left[ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)+i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right]}^{n}}\] \[{{\left[ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)+i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right]}^{n}}\] \[={{\left[ \cos \frac{1}{2}\left( \frac{\pi }{2}-\theta \right)+i\sin \frac{1}{2}\left( \frac{\pi }{2}-\theta \right) \right]}^{2n}}\] \[=\cos n\left( \frac{\pi }{2}-\theta \right)+i\sin n\left( \frac{\pi }{2}-\theta \right)\]
You need to login to perform this action.
You will be redirected in
3 sec
