# Given w=2(cos150∘+isin150∘)w=2(cos150) and z=–sqrt{2}+i Find the polar form of z

Question
Given $$w=2(\cos150∘+i\sin150∘)w=2(\cos150) and z=–\sqrt{2}+i$$
Find the polar form of z

2021-03-03

We don't actually need ww in this case.
Remember that the polar form of a complex number is
$$z=re^{iθ}\oslash$$
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
$$r = \sqrt{√2^{2}+1^{2}}=\sqrt{3}$$
and that
$$\tan(π−\oslash) = \frac{1}{\sqrt{2}}$$
$$π−\oslash=\arctan \frac{1}{\sqrt{2}}$$
$$\oslash=\pi−\arctan \frac{1}{\sqrt{2}}$$
Hence the polar form of z is
$$\boxed{z = \sqrt{3}e^{[\pi - \arctan(1/,\sqrt{2})]i}}$$
which is approximately
$$z=1.73e^{2.53i}$$

### Relevant Questions

Given $$\displaystyle{w}={2}{\left({\cos{{150}}}∘+{i}{\sin{{150}}}∘\right)}{w}={2}{\left({\cos{{150}}}\right)}{\quad\text{and}\quad}{z}=–\sqrt{{{2}}}+{i}$$
Find the polar form of z
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