Question

Rectangular Coordinates to Polar Coordinates A point is graphed in rectangular form. Find polar coordinates for the point, with$r>0$and $0<\theta <2\pi$.

Short answer

The polar coordinates of the point $Q(0,-3)$are$3,\frac{3\pi }{2}$ .

Step-by-step solution

Step 1. Given information.

The rectangular coordinates of the point Q are $(0,-3)$.

Step 2. We can find the polar coordinates of the point using the rectangular to polar coordinates conversion formulas.

From the given figure, we know that the coordinates of the point Q are$(0,-3)$.

We can use the conversion formulas$r^2=x^2+y^2$and

$\tan \theta =\frac{y}{x}$ $\because {\sin }^{2}x+{\cos }^{2}x=1$

to find the polar coordinates. The value of x is 0 and the value of y is$-3$.$\begin{array}{l}\Rightarrow r^2=x^2+y^2\\ \Rightarrow r^2=\text{Invalid <m:msup> element}{0}^2+\\ \Rightarrow r^2=9\\ \Rightarrow r=\pm 3\\ \Rightarrow Sincer\ge 0,\end{array}$

the value of r is$+3$.

Now, to find the value of $\theta$,

$\begin{array}{l}\Rightarrow \tan \theta =\frac{y}{x}\\ \Rightarrow \tan \theta =\frac{-3}{0}\\ \Rightarrow \theta ={\tan }^{-1}(-\infty )\\ \Rightarrow \theta =\frac{3\pi }{2}\end{array}$