Amplitude modulus form of 1+sinx +icosx

Question

sillybilly123 · Answer

No of ways ways to do this

$ 1+\sin x+i\cos x $
 
$=  \sin {\pi \over 2} +\sin x +i(  \cos {\pi \over 2}  + \cos x) $
 
thus setting up use of sum-product formulae.....
 
$= 2 \cos ({\pi \over 4} - {x \over 2})  \sin ({\pi \over 4} + {x \over 2}) +  2 i(  \cos ({\pi \over 4} + {x \over 2})  \cos ({\pi \over 4} - {x \over 2})) $

$= 2 \cos ({\pi \over 4} - {x \over 2})  \left( \sin ({\pi \over 4} + {x \over 2}) +   i(  \cos ({\pi \over 4} + {x \over 2})   \right)$
 
finally, switch cos and sine around in  bracketed part by using $ \sin  \theta = \cos ( \theta - {\pi \over 2} )$ and $ \cos  \theta = \sin ( \theta - {\pi \over 2} )$....

and so use Euler's formula for that complex bit to finish it off.

Ultrilliam · Answer

@Tejas