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STD 11 - 4.1 complex nubers — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 4.1 complex nubers4 Marks
MCQ
${\left( {\frac{{\cos \theta + i\sin \theta }}{{\sin \theta + i\cos \theta }}} \right)^4}$equals
  • A
    $\sin 8\theta - i\cos 8\theta $
  • B
    $\cos 8\theta - i\sin 8\theta $
  • C
    $\sin 8\theta + i\cos 8\theta $
  • ✓
    $\cos 8\theta + i\sin 8\theta $
✓

Answer

Correct option: D.
$\cos 8\theta + i\sin 8\theta $
d
(d) ${\left( {\frac{{\cos \,\theta + i\sin \theta }}{{\sin \,\theta + i\cos \,\,\theta }}} \right)^4} = \frac{{{{(\cos \,\,\theta + i\sin \,\theta )}^4}}}{{{i^4}\,{{(\cos \,\,\theta - i\sin \theta )}^4}}}$$ = \frac{{\cos \,\,4\theta + i\sin \,\,4\theta }}{{\cos \,\,4\theta - i\sin \,\,4\theta }}$
$ = \frac{{(\cos 4\theta + i\sin 4\theta )(\cos 4\theta + i\sin 4\theta )}}{{(\cos 4\theta - i\sin 4\theta )(\cos 4\theta + i\sin 4\theta )}}$
$\frac{{{{(\cos 4\theta + i\sin 4\theta )}^2}}}{{{{\cos }^2}4\theta + {{\sin }^2}4\theta }} = \cos 8\theta + i\sin 8\theta $

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