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, \(\displaystyle{z}=-\frac{{5}}{\sqrt{{2}}}+\frac{{t}}{\sqrt{{2}}}{i},{z}=-\frac{{{5}\sqrt{{2}}}}{{2}}+\frac{{{5}\sqrt{{2}}}}{{2}}{i}\)

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, \(\displaystyle{z}=-\frac{{5}}{\sqrt{{2}}}+\frac{{t}}{\sqrt{{2}}}{i},{z}=-\frac{{{5}\sqrt{{2}}}}{{2}}+\frac{{{5}\sqrt{{2}}}}{{2}}{i}\)