Question

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with$r>0$ and$0\le \theta <2\pi .$$(3\sqrt{3},-3)$.

Short answer

The polar coordinates of the point $(3\sqrt{3},-3)$are$(6,\frac{11\pi }{6})$ .

Step-by-step solution

Step 1.Given information.

The polar coordinates of the point are$(3\sqrt{3},-3)$ .

Step 2. Since the rectangular coordinates of the point are (33,−3) we can say thatx=33,y=−3 .

$\begin{array}{l}y=-3\\ \Rightarrow r^2=x^2+y^2\\ \Rightarrow r^2={\left(3\sqrt{3}\right)}^2+{(-3)}^2\\ \Rightarrow r^2=27+9\\ \Rightarrow r^2=36\\ \Rightarrow r=\pm 6\end{array}$

Step3. Finding the value of angle

$\begin{array}{l}\Rightarrow \tan \theta =\frac{y}{x}\\ \Rightarrow \tan \theta =\frac{-3}{3\sqrt{3}}\\ \Rightarrow \theta ={\tan }^{-1}(-\frac{1}{\sqrt{3}})\\ \Rightarrow \theta =\frac{5\pi }{6},\frac{11\pi }{6}\text{inthegivenrange}0\le \theta \le 2\pi \end{array}$

For the given range $0\le \theta \le 2\pi$

Since the point$(3\sqrt{3},3)$ lies in the the fourth quadrant, we take the value of$\theta =\frac{11\pi }{6}$ .