Complex Numbers II: Modulus & Argument
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
Find the modulus and argument (in radians) of the following:
(a)\(z=3+3i\)
(b)\(z=-\sqrt{3}+i\)
(c)\(z=-2-2i\)
(a)
Modulus and argument
\(|z|=3\sqrt{2};\;rg(z)=rctan(1)= frac{\pi}{4}\)
Answer
\(|z|=3\sqrt{2},\;rg z=\dfrac{\pi}{4}\)
(b)
Q2: \(|z|=2\)
\(rg z=\pi-rctan frac{1}{\sqrt3}=\pi- frac{\pi}{6}= frac{5\pi}{6}\)
Answer
\(|z|=2,\;rg z=\dfrac{5\pi}{6}\)
(c)
Q3: \(|z|=2\sqrt{2}\)
\(rg z=-\pi+ frac{\pi}{4}=- frac{3\pi}{4}\)
Answer
\(|z|=2\sqrt{2},\;rg z=-\dfrac{3\pi}{4}\)
Question 2
Write the following in polar form \(r(\cos heta+i\sin heta)\):
(a)\(z=1+i\)
(b)\(z=-\sqrt{3}-i\)
(a)
Answer
\(\sqrt{2}\!\left(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4}
ight)\)
(b)
Q3: \(r=2,\; heta=- frac{5\pi}{6}\)
Answer
\(2\!\left(\cos\!\left(-\dfrac{5\pi}{6}
ight)+i\sin\!\left(-\dfrac{5\pi}{6}
ight)
ight)\)
Question 3
Convert from polar to Cartesian form:
(a)\(4\!\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}
ight)\)
(b)\(3\, ext{cis}\!\left(-\dfrac{\pi}{2}
ight)\)
(a)
Answer
\(4\!\left( frac{1}{2}+ frac{\sqrt3}{2}i
ight)=2+2\sqrt{3}i\)
Question 4
Given \(z_1=2\, ext{cis}\!\left(\dfrac{\pi}{6}
ight)\) and \(z_2=3\, ext{cis}\!\left(\dfrac{\pi}{4}
ight)\), find \(z_1z_2\) and \(\dfrac{z_1}{z_2}\) in polar form.
Multiplication and division rules
Multiply moduli, add/subtract arguments
Answer
\(z_1z_2=6\, ext{cis}\!\left(\dfrac{5\pi}{12}
ight)\); \(\dfrac{z_1}{z_2}=\dfrac{2}{3}\, ext{cis}\!\left(-\dfrac{\pi}{12}
ight)\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets