Exercise · ZW-POLAR-FORM-007 · ★★☆ Intermediate
What is the principal argument, in \((-\pi,\pi]\), of \(-2-2i\)?
Choose one option.
Show answer
- D. \(-\dfrac{3\pi}{4}\)
Explanation
The point is in the third quadrant, and \(-2-2i=2\sqrt2\left(\cos\left(-\frac{3\pi}4\right)+i\sin\left(-\frac{3\pi}4\right)\right)\). The value \(\pi/4\) comes from \(\arctan(1)\) ignoring the quadrant.
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-006 Let z=4(cos (5π)/(6)+isin (5π)/(6)) and w=2(cos π3+isin π3). Compute z/w.
- 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?