# Write standart form of complex number 5(cos135^@+isin135^@)

Complex numbers
Write standart form of complex number $$\displaystyle{5}{\left({{\cos{{135}}}^{\circ}+}{i}{\sin{{135}}}^{\circ}\right)}$$

2021-01-08
Standart form of complex number, $$\displaystyle{z}={r}{\left({\cos{\theta}}+{i}{\sin{\theta}}\right)}$$, is z = a+ib
Calculate the values of cosine and sine,
$$\displaystyle{\cos{{\left({135}^{\circ}\right)}}}=-\frac{{1}}{\sqrt{{2}}}$$
$$\displaystyle{\sin{{\left({135}^{\circ}=\frac{{1}}{\sqrt{{2}}}\right.}}}$$
Substitute the values od cosine and sine in $$\displaystyle{z}={5}{\left({{\cos{{135}}}^{\circ}+}{i}{\sin{{135}}}^{\circ}\right)},$$
$$\displaystyle{z}={5}{\left({{\cos{{135}}}^{\circ}+}{i}{\sin{{135}}}^{\circ}\right)}={5}{\left(-\frac{{1}}{\sqrt{{2}}}+{i}\frac{{1}}{\sqrt{{2}}}\right)}=-\frac{{5}}{\sqrt{{2}}}+\frac{{t}}{\sqrt{{2}}}{i}$$
Rationalise the complex number, $$\displaystyle{z}=-\frac{{5}}{\sqrt{{2}}}+\frac{{t}}{\sqrt{{2}}}{i},{z}=-\frac{{{5}\sqrt{{2}}}}{{2}}+\frac{{{5}\sqrt{{2}}}}{{2}}{i}$$