Question Type 5: Applying multiplication and division to complex numbers in polar form Exercises

Multiply the complex numbers 5igl(\, ext{cos} frac{ ext{π}}{16}+i ext{sin} frac{ ext{π}}{16}\bigr) and 3igl( ext{cos} frac{3 ext{π}}{16}+i ext{sin} frac{3 ext{π}}{16}\bigr) and express the answer in polar form.

Compute the product $4\bigl(\text{cos}\tfrac{2\pi}{3}+i\text{sin}\tfrac{2\pi}{3}\bigr)\times6\bigl(\text{cos}\tfrac{5\pi}{6}+i\text{sin}\tfrac{5\pi}{6}\bigr)$ and give the answer in polar form.

Divide $8\bigl(\text{cos}\tfrac{5\pi}{12}+i\text{sin}\tfrac{5\pi}{12}\bigr)$ by $2\bigl(\text{cos}\tfrac{\pi}{6}+i\text{sin}\tfrac{\pi}{6}\bigr)$ and express the result in polar form.

Find the product $7\bigl(\text{cos}\tfrac{7\pi}{8}+i\text{sin}\tfrac{7\pi}{8}\bigr)\times2\bigl(\text{cos}\tfrac{9\pi}{8}+i\text{sin}\tfrac{9\pi}{8}\bigr)$ in polar form.

Compute the product $5\bigl(\text{cos}\tfrac{\pi}{5}+i\text{sin}\tfrac{\pi}{5}\bigr)\times5\bigl(\text{cos}\tfrac{2\pi}{5}+i\text{sin}\tfrac{2\pi}{5}\bigr)$ and give the answer in polar form.

Divide $10\bigl(\text{cos}\tfrac{13\pi}{12}+i\text{sin}\tfrac{13\pi}{12}\bigr)$ by $5\bigl(\text{cos}\tfrac{7\pi}{12}+i\text{sin}\tfrac{7\pi}{12}\bigr)$ and express the result in polar form.

Express the quotient $\frac{8e^{i\frac{3\pi}{7}}}{4e^{i\frac{5\pi}{14}}}$ in the form $re^{i\theta}$.

Compute $\frac{3\bigl(\text{cos}\tfrac{\pi}{4}+i\text{sin}\tfrac{\pi}{4}\bigr)\times4\bigl(\text{cos}\tfrac{\pi}{3}+i\text{sin}\tfrac{\pi}{3}\bigr)}{2\bigl(\text{cos}\tfrac{\pi}{6}+i\text{sin}\tfrac{\pi}{6}\bigr)}$ and express the answer in polar form.

Express $\frac{5e^{i\frac{\pi}{5}}\times2e^{i\frac{3\pi}{10}}}{3e^{i\frac{\pi}{2}}}$ in the form $Re^{i\theta}$.

Compute $\biggl(\frac{9\bigl(\text{cos}\tfrac{4\pi}{5}+i\text{sin}\tfrac{4\pi}{5}\bigr)}{3\bigl(\text{cos}\tfrac{7\pi}{10}+i\text{sin}\tfrac{7\pi}{10}\bigr)}\biggr)\times2\bigl(\text{cos}\tfrac{\pi}{10}+i\text{sin}\tfrac{\pi}{10}\bigr)$ in polar form.

Evaluate $\frac{12\bigl(\text{cos}\tfrac{11\pi}{12}+i\text{sin}\tfrac{11\pi}{12}\bigr)\times5\bigl(\text{cos}\tfrac{\pi}{3}+i\text{sin}\tfrac{\pi}{3}\bigr)}{6\bigl(\text{cos}\tfrac{5\pi}{12}+i\text{sin}\tfrac{5\pi}{12}\bigr)}$ and express the answer in polar form.

Find the product of three complex numbers in polar form: $2\bigl(\text{cos}\tfrac{\pi}{8}+i\text{sin}\tfrac{\pi}{8}\bigr)\times3\bigl(\text{cos}\tfrac{\pi}{4}+i\text{sin}\tfrac{\pi}{4}\bigr)\times5\bigl(\text{cos}\tfrac{\pi}{16}+i\text{sin}\tfrac{\pi}{16}\bigr).$