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

Question
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

2020-11-09
We don't actually need ww in this case.
Remember that the polar form of a complex number is
$$\displaystyle{z}={r}{e}^{{{i}θ}}{o}{s}{l}{a}{s}{h}$$
where rr is the norm of z and θ angle it makes with the xx axis.
[graph]
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
$$\displaystyle{r}=\sqrt{{√{2}^{{{2}}}+{1}^{{{2}}}}}=\sqrt{{{3}}}$$
and that
$$\displaystyle{\tan{{\left(π−{o}{s}{l}{a}{s}{h}\right)}}}={\frac{{{1}}}{{\sqrt{{{2}}}}}}$$
$$\displaystyleπ−{o}{s}{l}{a}{s}{h}={\arctan{{\frac{{{1}}}{{\sqrt{{{2}}}}}}}}$$
$$\displaystyle{o}{s}{l}{a}{s}{h}=\pi−{\arctan{{\frac{{{1}}}{{\sqrt{{{2}}}}}}}}$$
Hence the polar form of z is
$$\displaystyle{b}\otimes{e}{d}{\left\lbrace{z}=\sqrt{{{3}}}{e}^{{{\left[\pi-{\arctan{{\left({1}&#{x}{2}{F},\sqrt{{{2}}}\right)}}}\right]}{i}}}\right\rbrace}$$
which is approximately
$$\displaystyle{z}={1.73}{e}^{{{2.53}{i}}}$$

### Relevant Questions

Given $$w=2(\cos150∘+i\sin150∘)w=2(\cos150) and z=–\sqrt{2}+i$$
Find the polar form of z
Let $$\displaystyle{f{{\left({x},{y}\right)}}}=-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}$$.
Find limit of $$\displaystyle{f{{\left({x},{y}\right)}}}{a}{s}{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}{i}{)}{A}{l}{o}{n}{g}{y}{a}\xi{s}{\quad\text{and}\quad}{i}{i}{)}{a}{l}{o}{n}{g}{t}{h}{e}{l}\in{e}{y}={x}.{E}{v}{a}{l}{u}{a}{t}{e}\Lim{e}{s}\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{y}.{\log{{\left({x}^{{2}}+{y}^{{2}}\right)}}}$$,by converting to polar coordinates.
1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”
Parametric to polar equations Find an equation of the following curve in polar coordinates and describe the curve. $$x = (1 + cos t) cos t, y = (1 + cos t) sin t, 0 \leq t \leq 2\pi$$
Investigation Consider the helix represented by the vector-valued function $$r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >$$ (a) Write the length of the arc son the helix as a function of t by evaluating the integral $$s=\ \int_{0}^{t}\ \sqrt{[x'(u)]^{2}\ +\ [y'(u)]^{2}\ +\ [z'(u)]^{2}\ du}$$
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=3\ \ln(t),\ y=4t^{\frac{1}{2}},\ z=t^{3},\ (0,\ 4,\ 1)$$
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=e^{-8t}\ \cos(8t),\ y=e^{-8t}\ \sin(8t),\ z=e^{-8t},\ (1,\ 0,\ 1)$$
Find the coordinates of the point on the helix for arc lengths $$s =\ \sqrt{5}\ and\ s = 4$$. Consider the helix represented investigation by the vector-valued function $$r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >$$.
Write a short paragraph explaining this statement. Use the following example and your answers Does the particle travel clockwise or anticlockwise around the circle? Find parametric equations if the particles moves in the opposite direction around the circle. The position of a particle is given by the parametric equations $$x = sin t, y = cos t$$ where 1 represents time. We know that the shape of the path of the particle is a circle.
Write a short paragraph explaining this statement. Use the following example and your answers How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. The position of a particle is given by the parametric equations $$x = sin t, y = cos t$$ where 1 represents time. We know that the shape of the path of the particle is a circle.