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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 ?

MuhammadJ
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1 Answers1

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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 $\theta = \frac{2}{3}\pi$.

brick
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  • The Question is : Why we choose the angle 120 but not the angle 60 ? I mean why we use the inverse of cos not the sin to get the angle? – MuhammadJ Nov 23 '14 at 21:41
  • @MuhammadJuma'a Well you need both $sin$ and $cos$. $sin(120^0) = sin(60^o)$, and you don't know which angle to pick. Here comes $cos$ to help you. $cos(120^o) = -\frac{1}{2}$, while $cos(60^o) = \frac{1}{2}$. So now you know that the answer is $120^o$. For every $0 \neq x \in (-1, 1)$ you have $2$ values $\theta \in [0, 2\pi)$, for which $sin(\theta) = x$. Same with $cos$. – brick Nov 23 '14 at 21:57