Question

Express the following numbers in the form A+iB : ​

Answer 1

Answer:

$- \frac{a}{2} ( - 1 + i \cot( \frac{ \alpha }{2} ))$

Step-by-step explanation:

$\frac{a}{1 - \cos( \alpha ) - i \sin( \alpha ) } = \frac{a}{2 { \sin( \frac{ \alpha }{2} ) }^{2} - 2i \sin( \frac{ \alpha }{2} ) \cos( \frac{ \alpha }{2} ) ) } \\ \\ = \frac{a}{2 \sin( \frac{ \alpha }{2} )( \sin( \frac{ \alpha }{2} ) - i \cos( \frac{ \alpha }{2} ) ) } \\ \\ = \frac{a}{2i \sin( \frac{ \alpha }{2} )( \cos( \frac{ \alpha }{2} ) - i \sin( \frac{ \alpha }{2} ) ) } \\ \\ = \frac{ - ai( \cos( \frac{ \alpha }{2} ) + i \sin( \frac{ \alpha }{2} ) )}{2 \sin( \frac{ \alpha }{2} )} \\ \\ = - \frac{ai}{2} \frac{( \cot( \frac{ \alpha }{2} ) + i )}{1} \\ \\ = - \frac{a}{2} ( - 1 + i \cot( \frac{ \alpha }{2} ))$