6.3 Polar form of complex numbers

If [math]-\sqrt{3}-3i=\sqrt{a}[/math] cis[math]\theta[/math] where [math]a[/math] is an integer and [math]0\leq\theta<2\pi[/math], what are [math]a[/math] and [math]\theta[/math]?

If [math]1-i=\sqrt{a}[/math] cis[math]\theta[/math] where [math]a[/math] is an integer and [math]0\leq\theta<2\pi[/math], what are [math]a[/math] and [math]\theta[/math]?

Rewrite the following numbers in polar form:

(a) [math]\cos\left(\displaystyle{\frac{\pi}{7}}\right)-i\sin\left(\displaystyle{\frac{\pi}{7}}\right)[/math]

(b) [math]-3\left(\cos\dfrac{2\pi}{5}+i\sin\dfrac{2\pi}{5}\right)[/math]

(c) [math]4\left(\sin\dfrac{\pi}{8}+i\cos\dfrac{\pi}{8}\right)[/math]

(d) [math]15i[/math]

(e) [math]-6[/math]

What is [math]\arg(e^{3-5i})[/math]?

Let [math]z_1[/math] and [math]z_2[/math] be complex numbers satisfying, [math]|z_1|=2, |z_2|=5, \arg(z_1)=\dfrac{\pi}{3}[/math], and [math]\arg(z_2)=\dfrac{3\pi}{4}[/math]. Evaluate:

(a) [math]z_1z_2[/math]

(b) [math]\dfrac{z_1}{z_2}[/math]

(c) [math](z_2+z^*_2)^2[/math]

Square [math]ABCD[/math] is on the complex plane. Suppose [math]A=-1+4i[/math] and [math]B=-3[/math]. Find [math]C[/math] and [math]D[/math].

Let [math]A, B, C\in\mathbb{C}[/math] satisfying [math]|A|=|B|=|C|[/math]. [math]A=\sqrt{2}+\sqrt{2}i[/math] and these numbers form an equilateral triangle. Given that [math]\arg{B}>\arg{C}[/math], find [math]B[/math] and [math]C[/math].

What are the modulus and argument of the complex number [math]\dfrac{1}{r(\cos\theta+i\sin\theta)}=\dfrac{1}{r}(\cos(-\theta)+i\sin(-\theta))[/math]?

Let [math]z=5(\cos\theta+i\sin\theta)[/math] where [math]\theta[/math] is in the second quadrant. If [math]\cos\theta=-\dfrac{3}{5}[/math]. Find [math]\arg((z-1)^2)[/math]

Evaluate [math]\displaystyle{\sum^{2024}_{k=1}e^{\frac{ik\pi}{2}}}[/math].

Let [math]z[/math] be a complex number satisfying [math]z(\sqrt{3}+i)=-2\sqrt{3}+2i[/math]. Find [math]\arg(z)[/math].

Find the arguments of the following complex numbers. Express your answers in integers or fractions.

(a) [math]-6i[/math]

(b) [math]\sqrt{2}-\sqrt{6}i[/math]

(c) [math]3\left(\sin\dfrac{\pi}{7}-i\cos\dfrac{\pi}{7}\right)[/math]

(d) [math]-3\left(\cos\dfrac{3\pi}{8}+i\sin\dfrac{3\pi}{8}\right)[/math]

(e) [math]e^{4+3i}[/math]

(f) [math]e^{20}[/math]

Let [math]z_1[/math] and [math]z_2[/math] be complex numbers satisfying [math]i^3z_1=2z^*_2[/math]. Suppose [math]|z_1|=3[/math] and [math]\arg(z_1)=\dfrac{3\pi}{5}[/math]. Find [math]z_2[/math].

[math]A=z_1, B=z_2[/math], and [math]C=z_3[/math] are three complex numbers on the plane. Given that [math]ABC[/math] is a right isosceles triangle and [math]\angle B=90^\circ[/math],
(a) show that [math](z_1-z_2)^2=-(z_2-z_3)^2[/math].
(b) find [math]D[/math] such that [math]ABCD[/math] is a square.