IB Mathematics Applications & Interpretation Question Type 2: Converting Complex Numbers from Euler to Polar Form Exercises

Express $z$ in the form $r(\cos \theta +i\sin \theta )$, where $r>0$ and $0\le \theta <2\pi$.

Write $7{\mathrm{e}}^{-\mathrm{i}5\pi /2}$ in polar form with a principal argument in $(-\pi ,\pi ]$.

Convert $9{\text{e}}^{\text{i}\frac{13\pi }{6}}$ to $r(\cos \theta +\text{i}\sin \theta )$ form with the argument in $(-\pi ,\pi ]$.

Convert $6e^{i\frac{10\pi }{3}}$ to polar form by first reducing the argument to $[0,2\pi )$.

Write $14e^{i17\pi /5}$ in trigonometric form, reducing the argument to the interval $[0,2\pi )$.

Convert $5e^{-i\frac{\pi }{3}}$ to polar form $r(\cos \theta +i\sin \theta )$.

Express $8e^{i\left(-\frac{3\pi }{4}\right)}$ in the polar form $r(\cos \theta +i\sin \theta )$, where $0\le \theta <2\pi$.

Write $10e^{i15\pi /4}$ in polar form by reducing its argument to the range $(-\pi ,\pi ]$.

Express $2e^{i\frac{7\pi }{4}}$ in polar form with the principal argument in the interval $(-\pi ,\pi ]$.

Express $4e^{i\frac{\pi }{4}}$ in polar form $r(\cos \theta +i\sin \theta )$.

Convert $5e^{i\left(-\frac{13\pi }{4}\right)}$ to $r(\cos \theta +i\sin \theta )$ form, giving the principal argument in the interval $(-\pi ,\pi ]$.

Write $3{\text{e}}^{\text{i}\frac{5\pi }{6}}$ in the form $r(\cos \theta +\text{i}\sin \theta )$.