Exercise · ZW-POLAR-FORM-006 · ★★☆ Intermediate
Let \(z=4\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)\) and \(w=2\left(\cos\frac\pi3+i\sin\frac\pi3\right)\). Compute \(z/w\).
Choose one option.
Show answer
- B. \(2i\)
Explanation
In a quotient, moduli divide (\(4/2=2\)) and arguments subtract (\(\frac{5\pi}6-\frac\pi3=\frac\pi2\)): \(2i\). Multiplying moduli and adding arguments gives the distractor \(-4\sqrt3-4i\).
Review the concept: Polar form
The polar form of a complex number \(z\neq0\) expresses it with its modulus \(r\) and an argument \(\theta\), an angle that \(z\) makes with the positive real axis.
\[z=r(\cos\theta+i\sin\theta),\qquad r=|z|\]
More practice problems: polar form
- ZW-POLAR-FORM-001 What is the polar form of 1+i?
- ZW-POLAR-FORM-002 Write -1 in polar form with θ∈(-π,π].
- ZW-POLAR-FORM-003 What is the principal argument of -√3+i?
- ZW-POLAR-FORM-004 Write 2(cos π3+isin π3) in the form a+bi.
- ZW-POLAR-FORM-005 Compute 2(cos π6+isin π6)·3(cos π3+isin π3).
- ZW-POLAR-FORM-007 What is the principal argument, in (-π,π], of -2-2i?
- ZW-POLAR-FORM-008 A complex number has argument π/4. Which of these is also an argument of it?
- ZW-POLAR-FORM-009 Why is the polar form defined only for z≠0?