Question

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given.$(4,\pi /6)$

Short answer

The rectangular coordinates of the point $(4,\frac{\pi }{6})$are.${(2\sqrt{3},2)}^{.}$

Step-by-step solution

Step 1.Given information.

The polar coordinates of the point are$(4,\frac{\pi }{6})$ .

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

Since the polar coordinates are $(4,\frac{\pi }{6})$, we can say that $r=4$and

$\begin{array}{l}\theta =\frac{\pi }{6}\\ \Rightarrow x=r\cos \theta \\ \Rightarrow x=4\cos \left(\frac{\pi }{6}\right)\\ \Rightarrow x=4\cdot \left(\frac{\sqrt{3}}{2}\right)\\ \Rightarrow x=2\sqrt{3}\end{array}$

Now, to find the value of y,

$\begin{array}{l}\Rightarrow y=r\sin \theta \\ \Rightarrow y=4\sin \left(\frac{\pi }{6}\right)\\ \Rightarrow y=4\cdot \left(\frac{1}{2}\right)\\ \Rightarrow y=2\end{array}$