What are the polar coordinates of (2√3, 2)?

[onesun0000]

My answer to this is [MATH](4, π/6)[/MATH]. But a calculator said that [MATH](−4, 7π/6)[/MATH] is also an answer. Are both correct answers?

 

[pka]

My answer to this is [MATH](4, π/6)[/MATH]. But a calculator said that [MATH](−4, 7π/6)[/MATH] is also an answer. Are both correct answers?

There is no hard and fast rule. But it is my preference that yours is the one that I would expect if I were your lecturer.
The reason being that the form \(\displaystyle (r,\theta)\) where \(\displaystyle r>0~\&~-\pi<\theta\le \pi\). You see \(\displaystyle r\) is the absolute value of the number and \(\displaystyle \theta\) its identifies location.
Note some authors insist that \(\displaystyle 0\le\theta<2\pi\).

 

[onesun0000]

I saw this explanation in an Algebra II book. (I don't know the book name because I just saw the polar coordinates section).


It gave two answers. I don't understand why those two can be both polar coordinates of (-1,1).

 

[Dr.Peterson]

You can start at the origin, face in the 135 degree direction and move forward 1.414 units, or you can face in the -45 degree direction and move backward 1.414 units; you will end up in the same place. So although the second answer would typically be preferred, both are valid descriptions of the same point. In fact, there are other pairs you could give, using coterminal angles. Unlike Cartesian coordinates, there are multiple "names" for the same point.

 

[pka]

I saw this explanation in an Algebra II book. (I don't know the book name because I just saw the polar coordinates section).
It gave two answers. I don't understand why those two can be both polar coordinates of (-1,1).

This is the most important question: "What does your textbook/instructor say about polar coordinates?"
As I said above I use the range \(\displaystyle -\pi<\theta\le\pi\) you may find this useful. For any point \(\displaystyle (x,y)\) where \(\displaystyle x\cdot y\ne 0\)
\(\displaystyle \arg(x , y) = \left\{ {\begin{array}{{ll}} {\arctan \left( {\frac{y}{x}} \right),}&{x > 0} \\ {\arctan \left( {\frac{y}{x}} \right) + \pi ,}&{x < 0\;\& \;y > 0} \\ {\arctan \left( {\frac{y}{x}} \right) - \pi ,}&{x < 0\;\& \;y < 0} \end{array}} \right. \)

If \(\displaystyle (x,y)\) is on a major axis you should know what \(\displaystyle \theta\) is.

Here is a very good web-page

 

[onesun0000]

I now understand this. Thank you so much.

 

[topsquark]

I have never seen anyone use r < 0. That's just bizarre.

-Dan

 

[Dr.Peterson]

As indicated in the problem, the point is not that one would give the negative r as an answer to a question asking for the polar coordinates of a point, but that such a pair can be plotted, and is valid. It shows up, in particular, in graphs of polar equations, where for example some lobes may be formed by negative values of r. For example, see examples 1 and 2 here. (Example 3 is almost my logo, which doesn't involve negative r.)