Question

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with $r<0$and the other with$r>0$$(-2,-\frac{\pi }{3})$

Short answer

The graph of a given point,

The other two points are,$(-2,\frac{5\pi }{3})$$(2,\frac{2\pi }{3})$

Step-by-step solution

Step 1. Given information

A point in polar coordinates as $(-2,-\frac{\pi }{3})$

Step 2. Concept used

The point is plotted or presented on the graph using the polar coordinate system. The first coordinate represents the radius, while the second coordinate represents the angle.

Step 3. Calculation

Now plot the point on the graph,

Now for$r>o$, add into$2\pi$the coordinate-

$\begin{array}{l}\Rightarrow (-2,-\frac{\pi }{3})=(-2,-\frac{\pi }{3}+2\pi )\\ \Rightarrow (-2,-\frac{\pi }{3})=(-2,\frac{5\pi }{3})\end{array}$

For $r<o$, change the sign of radius and add $\pi$to angle given-

$\begin{array}{l}\Rightarrow (-2,-\frac{\pi }{3})=\left(-(-2),-\frac{\pi }{3}+\pi \right)\\ \Rightarrow (-2,-\frac{\pi }{3})=(2,\frac{2\pi }{3})\end{array}$