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4-1.Complex numbers
easy
$\frac{1}{{1 - \cos \theta + i\,\sin \theta }}$ નો વાસ્તવિક ભાગ મેળવો.
A
$1/4$
B
$1/2$
C
$tan {\theta\over2}$
D
$1/{1-cos\theta }$
Solution
(b) $\frac{1}{{1 – \cos \,\,\theta + i\sin \,\,\theta }}$$ = \frac{1}{{(1 – \cos \,\theta ) + i\sin \,\theta }} \times \frac{{(1 – \cos \,\theta ) – i\sin \,\theta }}{{(1 – \cos \,\theta ) – i\sin \,\theta }}$
$ = \frac{{(1 – \cos \theta ) – i\sin \theta }}{{{{(1 – \cos \theta )}^2} + {{\sin }^2}\theta }}$ $ = \frac{{(1 – \cos \theta ) – i\sin \theta }}{{2(1 – \cos \theta )}}$
$ = \frac{{(1 – \cos \theta )}}{{2(1 – \cos \theta )}} – i\,\frac{{\sin \theta }}{{2(1 – \cos \theta )}}.$
Therefore its real part = $\frac{{1 – \cos \theta }}{{2\,(1 – \cos \theta )}} = \frac{1}{2}$
Standard 11
Mathematics
