Question

The complex number $e^{i\theta }$ can be expressed in vector form by

• A. $\sin \theta +i\cos \theta$
• B. $\cos \theta +i\sin \theta$
• C. both $\left(a\right)$ and $\left(b\right)$
• D. none of these

Answer 1

Answer: (B)

$\cos \theta +i\sin \theta$

Let $f\left(x\right)=\cos x+i\sin x$

Taking derivative we get,

$f^{\prime }\left(x\right)=-\sin x+i\cos x=i\times f\left(x\right)$

Any other function $g\left(x\right)$ has this property if

$g^{\prime }\left(x\right)=ig\left(x\right)$ i.e.

$\frac{dg}{dx}=ig$

$⟹\frac{1}{g}dg=idx$

Taking integration on both sides we get,

$log\left(g\right)=ix+C$

$⟹g=e^{(ix+C)}$

$⟹g=e^{ix}{e}^C$

$⟹g\left(x\right)=e^{ix}{c}_1$

Put $x=0$ we get $c_1=g\left(0\right)$

This function is $1$ when $c_1=1$

Hence, $\cos x+i\sin x=e^{ix}$