Choosing the angle in rectangular coordinates

Question

Find all possible polar coordinates for the point P that has rectangular coordinates ( -2,2 (3)^(1/2) ). At the end, the equation satisfied by which angle ? How to know it ? The cos angle or the sin angle ?

Answer 1

Polar coordinates of a point $(x,y)$ are given by distance $d$ from $(0,0)$ to the point, and an angle $\theta$ from $Ox$ to the vector $(x, y)$, measured in positive direction. In your case $d = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = 4$. To calculate the angle it's easier to make the length of the vector $1$, so that if you take its beginning to be $(0,0)$, its end will lie on the unit circle. Then from the definition of $sin$ and $cos$, its first coordinate will be $cos(\theta)$ and the second coordinate : $sin(\theta)$. So the unit vector in your case is $\frac{1}{d}(-2, 2\sqrt{3}) = (-\frac{1}{2}, \frac{\sqrt{3}}{2})$. Therefore $cos(\theta) = -\frac{1}{2}$ and $sin(\theta) = \frac{\sqrt{3}}{2}$. From where you can conclude that $\theta = \frac{2}{3}\pi$.