Question

Compute.
$\frac{4(\cos {75}^{\circ }+i\sin {75}^{\circ })}{0.4(\cos {30}^{\circ }+i\sin {30}^{\circ })}$

Answer 1

$\frac{4(\cos {75}^{\circ }+i\sin {75}^{\circ })}{0.4(\cos {30}^{\circ }+i\sin {30}^{\circ })}$

$=\frac{10(\cos {75}^{\circ }+i\sin {75}^{\circ })(\cos {30}^{\circ }-i\sin {30}^{\circ })}{{\cos }^2{30}^{\circ }-i^2{\sin }^2{30}^{\circ }}$

$=\frac{10\left[\cos \right({75}^{\circ }-{30}^{\circ })+i\sin ({75}^{\circ }-{30}^{\circ }\left)\right]}{{\cos }^2{30}^{\circ }+{\sin }^2{30}^{\circ }}$

$=10[\cos {45}^{\circ }+i\sin {45}^{\circ }]=\frac{10}{\sqrt{\left(2\right)}}(1+i)$

or $10\frac{e^{i\theta }}{e^{i\theta }}=10e^{i(\theta -\varphi )}=10e^{i\pi /4}=\frac{10}{\sqrt{\left(2\right)}}(1+i)$