Question
Download Solution PDF(sin x + icos x)3 जहाँ \(i=\sqrt{-1}\) का वास्तविक भाग क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFधारणा:
सम्मिश्र संख्याओं पर यूलर का सूत्र:
- eix = cos x + i sin x
- e-ix = cos x - i sin x
गणना:
(sin x + icos x)3
I को उभयनिष्ठ लें, हमें मिलता है
(sin x + icos x)3
= \({{\rm{i}}^3}{\left( {\frac{{\sin {\rm{x}}}}{{\rm{i}}} + {\rm{\;}}\cos {\rm{x}}} \right)^3}\)
= -i × (-i sin x + cos x)3
(∵ i3 = -i और 1/i = -i)
= -i × (cos x - i sin x) 3
\(= {\rm{}} - {\rm{i\;}} \times {\rm{}}{\left( {{{\rm{e}}^{ - {\rm{ix}}}}} \right)^3} \)
\(= {\rm{}} - {\rm{i\;}} \times {\rm{\;}}{{\rm{e}}^{ - {\rm{i}}3{\rm{x}}}}{\rm{\;}}\)
(∵e-ix = cos x - i sin x)
= -i (cos 3x – i sin 3x)
= (-i cos 3x + i2 sin 3x)
= -sin3x – i cos 3x
∴ वास्तविक भाग = -sin 3x
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