Explained: "Given w=2(cos⁡150∘+isin⁡150∘)w=2(cos⁡150)andz=–2+i Find the polar form of z"

College
Calculus and Analysis
Integral Calculus
Parametric equations, polar coordinates, and vector-valued functions

sjeikdom0

Answered question

2020-11-08

Given

w = 2 (\cos 150 \circ + i \sin 150 \circ) w = 2 (\cos 150) and z = - \sqrt{2} + i

Find the polar form of z

Answer & Explanation

Cullen

Skilled2020-11-09Added 89 answers

We don't actually need ww in this case.
Remember that the polar form of a complex number is
$z = r e^{i θ}$
where rr is the norm of z and θ angle it makes with the xx axis.

Using a little trigonometry (form a right triangle using the negative xx axis, the arrow shown and a straight line segment between them), we know that
$r = \sqrt{\sqrt 2^{2} + 1^{2}} = \sqrt{3}$
and that
$\tan (π - o s l a s h) = \frac{1}{\sqrt{2}}$
$π - o s l a s h = \arctan \frac{1}{\sqrt{2}}$
$o s l a s h = π - \arctan \frac{1}{\sqrt{2}}$
Hence the polar form of z is
$\begin{matrix} z = \sqrt{3} e^{[π - \arctan (\frac{1}{\sqrt{2}})] i} \end{matrix}$
which is approximately
$z = 1.73 e^{2.53 i}$

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