Rectangular-to-Polar Point Conversion Find polar coordinates of the point whose rectangular coordinates are (-1, √3).

Question

Rectangular-to-Polar Point Conversion

Find polar coordinates of the point whose rectangular coordinates are [latex](-1, \sqrt{3})[/latex].

Accepted Answer

We begin with [latex](x, y) =(-1, \sqrt{3})[/latex] and use our three-step procedure to find a set of polar coordinates (r, θ).

Step 1 Plot the point (x, y). The point [latex](-1, \sqrt{3})[/latex] is plotted in quadrant II in Figure 6.26.

Step 2 Find r by computing the distance from the origin to (x, y).

[latex]r=\sqrt{x^2+y^2}=\sqrt{(-1)^2+(\sqrt{3})^2}=\sqrt{1+3}=\sqrt{4}=2[/latex]

Step 3 Find θ using [latex] an θ =\frac{y}{x}[/latex] with the terminal side of θ passing through (x, y).

[latex] an θ =\frac{y}{x}=\frac{\sqrt{3}}{-1}=-\sqrt{3}[/latex]

We know that [latex] an \frac{π}{3}=\sqrt{3}[/latex]. Because θ lies in quadrant II,

[latex]θ = π -\frac{π}{3}=\frac{3π}{3}-\frac{π}{3}=\frac{2π}{3}[/latex].

One representation of [latex](-1, \sqrt{3})[/latex] in polar coordinates is [latex](r, θ) = (2, \frac{2π}{3})[/latex].

Question 6.3.4: Rectangular-to-Polar Point Conversion Find polar coordinates...