Question

convert the following point from polar to cartesian coordinates. write the exact answer as an ordered pair. do not round. (3, -\frac{11\pi}{6})

Explanation:

Step1: Recall the conversion formulas

For a polar point \((r,\theta)\), the Cartesian coordinates \((x,y)\) are given by \(x = r\cos\theta\) and \(y=r\sin\theta\). Here \(r = 3\) and \(\theta=-\frac{11\pi}{6}\).

Step2: Find the \(x\) - coordinate

Use the formula \(x = r\cos\theta\).
Since \(\cos(-\frac{11\pi}{6})=\cos(\frac{11\pi}{6})\) (because \(\cos(-\alpha)=\cos\alpha\)), and \(\cos(\frac{11\pi}{6})=\cos(2\pi-\frac{\pi}{6})=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\).
So \(x = 3\times\frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{2}\).

Step3: Find the \(y\) - coordinate

Use the formula \(y = r\sin\theta\).
Since \(\sin(-\frac{11\pi}{6})=-\sin(\frac{11\pi}{6})\) (because \(\sin(-\alpha)=-\sin\alpha\)), and \(\sin(\frac{11\pi}{6})=\sin(2\pi - \frac{\pi}{6})=-\sin\frac{\pi}{6}=-\frac{1}{2}\).
So \(y=3\times(- (-\frac{1}{2}))=\frac{3}{2}\).

Answer:

\((\frac{3\sqrt{3}}{2},\frac{3}{2})\)