Question

Amplitude modulus form of 1+sinx +icosx

Answer 1

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.

Answer 2

@Tejas