We can represent the position of a point in the plane not only by its Cartesian coordinates [math]\left(x,y\right)[/math], but also by its polar coordinates [math]\left(r,\theta\right)[/math].[br][br]Interact with the app below to explore the relationship between the two coordinate systems.
It is customary to assign to the polar axis (corresponding to [math]\theta=0[/math]) the same direction of the [math]x[/math]-axis of an orthogonal Cartesian system, and to make the pole coincide with the origin, in order to simplify the conversion between the two systems.[br][br]With this convention, the [math]y[/math]-axis of the Cartesian system corresponds to [math]\theta=\frac{\pi}{2}[/math] in the polar system.[br]Polar functions will not have the Cartesian form [math]y=f\left(x\right)[/math], but instead the polar form [math]r=f\left(\theta\right)[/math].[br]
Determine the inverse formulas of the conversion formulas described in the app.[br]The inverse formulas allow you to obtain the polar coordinates [math]\left(r,\theta\right)[/math] of a point whose Cartesian coordinates [math]\left(x,y\right)[/math] are known.[br]Use these formulas to calculate the polar coordinates of the point whose Cartesian coordinates are [math]\left(-2,2\sqrt{3}\right)[/math].
The coordinates of the following points are given in polar form. [br][math]\left(4,\frac{\pi}{4}\right),\left(1,-\frac{\pi}{2}\right),\left(6,-\frac{2}{3}\pi\right)[/math] and [math]\left(6,\frac{4}{3}\pi\right)[/math].[br]Plot these points in a polar coordinate system. [br]Do you notice anything?