Complex Numbers And Quadratic Equations question 227
Question: $ \frac{{{(\cos \alpha +i\sin \alpha )}^{4}}}{{{(\sin \beta +i\cos \beta )}^{5}}}= $ [RPET 2002]
Options:
A) $ \cos (4\alpha +5\beta )+i\sin (4\alpha +5\beta ) $
B) $ \cos (4\alpha +5\beta )-i\sin (4\alpha +5\beta ) $
C) $ \sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta ) $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
$ \frac{{{(\cos \alpha +i\sin \alpha )}^{4}}}{{{(\sin \beta +i\cos \beta )}^{5}}} $ $ =\frac{\cos 4\alpha +i\sin 4\alpha }{i^{5}{{(\cos \beta -i\sin \beta )}^{5}}} $ = $ -i(\cos 4\alpha +i\sin 4\alpha ){{(\cos \beta -i\sin \beta )}^{-5}} $ = $ -i[\cos 4\alpha +i\sin 4\alpha ][\cos 5\beta +i\sin 5\beta ] $ = $ -i[\cos (4\alpha +5\beta )+i\sin (4\alpha +5\beta )] $ = $ \sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta ) $ .
BETA
