Show each step of your work and fully simplify each expression.
Turn in your answers in class on a physical piece of paper.
Staple multiple sheets together.
Feel free to use Desmos for graphing.
Answer the following:
Sketch a rough graph of the following:
$r = 2$
$\theta = \dfrac{\pi}{2}$
$r = 3\sin\theta$
$r = 6\cos \theta$
$\theta = -\dfrac{\pi}{4}$
What is the definition of a complex number?
Graph the following complex numbers and find their modulus:
$5 + 5i$
$-3 - 2i$
$i$
$\dfrac{-\sqrt{2} + i\sqrt{2}}{2}$
$\sqrt{3} + i$
Sketch the following sets in the complex plane:
$\left\{z = a + bi : a \leq 0, b \geq 0\right\}$
$\left\{z = a + bi : a \geq 1, b > 1\right\}$
$\left\{z : \lvert z \rvert = 3 \right\}$
$\left\{z : \lvert z \rvert \geq 1\right\}$
$\left\{z : 3 \leq \lvert z \rvert \leq 6 \right\}$
Write the complex number in polar form with argument $\theta$ between $0$ and $2\pi$.
$1 + i$
$- \sqrt{2} - \sqrt{2}i$
$-3 + 3i$
$1 + i$
$4$
$-2i$
$-3$
$-\sqrt{3} - i$
Given two complex numbers $z_1$ and $z_2$ in polar form, find $z_1z_2$ and $z_1/z_2$ in polar form.
$z_1 = 3 \left(\cos \dfrac{\pi}{3} + i \sin \dfrac{\pi}{3} \right)$ $z_2 = 2 \left(\cos \dfrac{\pi}{6} + i \sin \dfrac{\pi}{6}\right)$
$z_1 = \sqrt{3}\left(\cos \dfrac{5\pi}{6} + i \sin \dfrac{5\pi}{6}\right)$ $z_2 = 2(\cos \pi + i \sin \pi)$
$z_1 = 4 (\cos 120^\circ + i \sin 120^\circ)$ $z_2 = 2(\cos 30^\circ + i \sin 30^\circ)$
$z_1 = \sqrt{2}(\cos 75^\circ + i \sin75^\circ)$ $z_2 = 3\sqrt{2}(\cos 60^\circ + i \sin 60^\circ)$
Multiply the two complex numbers $z_1 = \sqrt{3} + i$ and $z_2 = 1 + \sqrt{3}i$ in two different ways:
Expanding the factors $z_1z_2$
Convert $z_1$ and $z_2$ to polar and multiply in polar. Also convert your answer back to $a + bi$ and check to see if it is the same numbers as in part (a).