Complex Numbers And Quadratic Equations question 698
Question: What is the real part of $ {{(\sin x+i\cos x)}^{3}} $ where $ i=\sqrt{-1} $ ?
Options:
A) $ -\cos 3x $
B) $ -\sin 3x $
C) $ \sin 3x $
D) $ \cos 3x $
Show Answer
Answer:
Correct Answer: B
Solution:
$ {{( sinx+icosx )}^{3}} $ $ = sin^{3}x + {{(i)}^{3}} cos^{3} x + 3i ( sin x ) ( cos x ) $ $ ( sin x + i cos x ) $ $ = sin x- i cos^{3} x + 3i sin^{2} x cos x- 3 sin x cos^{2} x $ $ =sin^{3} x-3sinxcos^{2} x+icosx( cos^{2} x+sin^{2} x ) $ $ =sinx( sin^{2} x-3cos^{2}x )+icosx $ Real part of $ {{( sin x + i cos x )}^{3}} $ $ = sin x ( sin^{2} x - 3 cos^{2} x ) $ $ =sinx[ sin^{2} x-3( 1-sin^{2}x ) ] $ $ = sin x [ 4 sin^{2}x - 3 ] $ $ =4 sin^{3} x-3sinx $ $ = -( 3 sin x - 4 sin^{3} x ) $ $ =-sin3x $
BETA
