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

Find the value of $\left( \frac{9(\cos \frac{4\pi}{5} + \mathrm{i}\sin \frac{4\pi}{5})}{3(\cos \frac{7\pi}{10} + \mathrm{i}\sin \frac{7\pi}{10})} \right) \times 2(\cos \frac{\pi}{10} + \mathrm{i}\sin \frac{\pi}{10})$ giving your answer in polar form.

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

Divide $8\left(\cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12}\right)$ by $2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)$ and express the result in polar form.

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

Compute the product $5\left(\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}\right)\times5\left(\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5}\right)$ and give the answer in polar form.

Divide $10\bigl(\cos\tfrac{13\pi}{12} + i \sin\tfrac{13\pi}{12}\bigr)$ by $5\bigl(\cos\tfrac{7\pi}{12} + i \sin\tfrac{7\pi}{12}\bigr)$ 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 the product $4\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)\times6\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$ and give 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 $Re^{i\theta}$.

Evaluate $\dfrac{12\left(\cos\frac{11\pi}{12}+i\sin\frac{11\pi}{12}\right)\times5\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)}{6\left(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\right)}$ and express the answer in polar form.

Multiply the complex numbers $5 \left( \cos \frac{\pi}{16} + i \sin \frac{\pi}{16} \right)$ and $3 \left( \cos \frac{3\pi}{16} + i \sin \frac{3\pi}{16} \right)$ and express the answer in polar form.

Find the product $7\left(\cos \frac{7\pi}{8} + i \sin \frac{7\pi}{8}\right) \times 2\left(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}\right)$ in polar form.