Question

find the rectangular coordinates for the point whose polar coordinates are given.
(3, 3π)
(x, y) = ( )

Explanation:

Step1: Recall polar to rectangular conversion formulas

The formulas to convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\) are \(x = r\cos\theta\) and \(y = r\sin\theta\). Here, \(r = 3\) and \(\theta = 3\pi\).

Step2: Calculate \(x\)-coordinate

Substitute \(r = 3\) and \(\theta = 3\pi\) into the formula for \(x\):
\(x = 3\cos(3\pi)\).
We know that \(\cos(3\pi)=\cos(\pi + 2\pi)=\cos(\pi)= - 1\) (since \(\cos\) has a period of \(2\pi\) and \(\cos(\pi)= - 1\)).
So, \(x = 3\times(-1)= - 3\).

Step3: Calculate \(y\)-coordinate

Substitute \(r = 3\) and \(\theta = 3\pi\) into the formula for \(y\):
\(y = 3\sin(3\pi)\).
We know that \(\sin(3\pi)=\sin(\pi + 2\pi)=\sin(\pi)=0\) (since \(\sin\) has a period of \(2\pi\) and \(\sin(\pi)=0\)).
So, \(y = 3\times0 = 0\).

Answer:

\((-3, 0)\)