Complex numbers › The complex plane

Argument and polar form

Solve the first problem 15 problems

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1. Polar form of a negative real number ●○○○○ 10 ptsWrite the complex number $-4$ in the form $r(\cos\theta + i\sin\theta)$ with $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $r$ and $\theta$, giving $\theta$ in degrees.
2. Polar form of a complex number ●○○○○ 10 ptsWrite $2-2i$ in polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-180^\circ<\theta\le 180^\circ$. Find $r$, rounded to two decimal places, and $\theta$ in degrees.
3. Principal value of i to the i ●●○○○ 20 ptsFind the principal value of $i^{i}$, where $i=\sqrt{-1}$. Round to two decimal places.
4. Argument of a sum of exponentials ●●○○○ 20 ptsThe number $e^{7\pi i/60} + e^{17\pi i/60} + e^{27\pi i/60} + e^{37\pi i/60} + e^{47\pi i/60}$ is written in the form $re^{i\theta}$, where $r \ge 0$ and $0 \le \theta < 2\pi$. Find $\theta$ in degrees.
5. Rotation center of a complex map ●●○○○ 20 ptsThe function $f(z) = \frac{(-1 + i \sqrt{3})\, z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$, writing $c = x + yi$ with $x$ and $y$ real. Round $x$ to two decimal places.
6. Simplify expression with imaginary cosine and sine ●●●○○ 50 ptsSimplify $2\cos^2(i\log 2009)+i\sin(i\log 4036081)$, where $\log$ is the natural logarithm. Give your answer as a decimal rounded to eight decimal places.
7. Argument of a sum of cis terms ●●●○○ 50 ptsThe number $\operatorname{cis} 75^\circ + \operatorname{cis} 83^\circ + \operatorname{cis} 91^\circ + \dots + \operatorname{cis} 147^\circ$ is expressed in the form $r \operatorname{cis} \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.

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