Question

the value of sin ix is​

Answer 1

Sinx =1 or -1.............

Answer 2

The value of sin ix is i sinh x

Given :

sin ix

To find :

The value

Solution :

Step 1 of 2 :

Write down expression for sin x

$\displaystyle \sf{ sin \: x = \frac{ {e}^{ix} - {e}^{ - ix} }{2i} }$

Step 2 of 2 :

Find the value of sin ix

$\displaystyle \sf{ sin \: ix = \frac{ {e}^{i.ix} - {e}^{ - i.ix} }{2i} }$

$\displaystyle \sf{ \implies sin \: ix = \frac{ {e}^{ {i}^{2} x} - {e}^{ - {i}^{2} x} }{2i} }$

$\displaystyle \sf{ \implies sin \: ix = \frac{ {e}^{ - x} - {e}^{ x} }{2i} }$

$\displaystyle \sf{ \implies sin \: ix = - \frac{ {e}^{ x} - {e}^{ - x} }{2i} }$

$\displaystyle \sf{ \implies sin \: ix = - i \frac{ {e}^{ x} - {e}^{ - x} }{2 {i}^{2} } }$

$\displaystyle \sf{ \implies sin \: ix = - i \frac{ {e}^{ x} - {e}^{ - x} }{ - 2 } }$

$\displaystyle \sf{ \implies sin \: ix = i .\frac{ {e}^{ x} - {e}^{ - x} }{ 2 } }$

$\displaystyle \sf{ \implies sin \: ix = i \: sin \: hx }$