Step 1

As a rule of thumb, if we add or subtract complex numbers, we will do that in rectangular form while multiplication, divison will be performed in polar form.

(a) \(\displaystyle{5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{\frac{{{3}\angle{60}^{{\circ}}}}{{{2}+{j}}}}\right)}={5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{\frac{{{3}\angle{60}^{{\circ}}}}{{{2.236}\angle{26.565}^{{\circ}}}}}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{1.3416}\angle{33.43}^{{\circ}}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{1.1196}+{j}{0.7392}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({7.13}-{j}{7.261}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({10.176}\angle-{45.52}^{{\circ}}\right)}\)

\(\displaystyle={50.88}\angle-{15.52}^{{\circ}}\)

Step 2

(b) \(\displaystyle{\frac{{{\left({10}\angle{60}^{{\circ}}\right)}{\left({35}\angle{60}^{{\circ}}\right)}}}{{{\left({2}+{j}{6}\right)}-{\left({5}+{j}\right)}}}}={\frac{{{\left({10}\angle{60}^{{\circ}}\right)}{\left({35}\angle-{50}^{{\circ}}\right)}}}{{-{3}+{j}{5}}}}\)

\(\displaystyle={\frac{{{\left({10}\angle{60}^{{\circ}}\right)}{\left({35}\angle-{50}^{{\circ}}\right)}}}{{{5.83}\angle{120.96}^{{\circ}}}}}\)

\(\displaystyle={60.02}\angle-{110.96}^{{\circ}}\)

As a rule of thumb, if we add or subtract complex numbers, we will do that in rectangular form while multiplication, divison will be performed in polar form.

(a) \(\displaystyle{5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{\frac{{{3}\angle{60}^{{\circ}}}}{{{2}+{j}}}}\right)}={5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{\frac{{{3}\angle{60}^{{\circ}}}}{{{2.236}\angle{26.565}^{{\circ}}}}}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{1.3416}\angle{33.43}^{{\circ}}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({6}-{j}{8}+{1.1196}+{j}{0.7392}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({7.13}-{j}{7.261}\right)}\)

\(\displaystyle={5}\angle{30}^{{\circ}}{\left({10.176}\angle-{45.52}^{{\circ}}\right)}\)

\(\displaystyle={50.88}\angle-{15.52}^{{\circ}}\)

Step 2

(b) \(\displaystyle{\frac{{{\left({10}\angle{60}^{{\circ}}\right)}{\left({35}\angle{60}^{{\circ}}\right)}}}{{{\left({2}+{j}{6}\right)}-{\left({5}+{j}\right)}}}}={\frac{{{\left({10}\angle{60}^{{\circ}}\right)}{\left({35}\angle-{50}^{{\circ}}\right)}}}{{-{3}+{j}{5}}}}\)

\(\displaystyle={\frac{{{\left({10}\angle{60}^{{\circ}}\right)}{\left({35}\angle-{50}^{{\circ}}\right)}}}{{{5.83}\angle{120.96}^{{\circ}}}}}\)

\(\displaystyle={60.02}\angle-{110.96}^{{\circ}}\)