Question

the polar coordinates of a point are $(-8,\frac{2\pi}{3})$. find the rectangular coordinates of this point. the rectangular coordinates are $\square$. (simplify your answer. type an ordered pair. type an exact answer for each coordinate, using radi

Explanation:

Step1: Recall the conversion formulas

The conversion formulas from polar coordinates \((r,\theta)\) to rectangular coordinates \((x,y)\) are \(x = r\cos\theta\) and \(y=r\sin\theta\). Here \(r=- 8\) and \(\theta=\frac{2\pi}{3}\).

Step2: Calculate the \(x\) - coordinate

We know that \(\cos\frac{2\pi}{3}=-\frac{1}{2}\). Using the formula \(x = r\cos\theta\), substitute \(r = - 8\) and \(\cos\theta=-\frac{1}{2}\). Then \(x=-8\times\cos\frac{2\pi}{3}=-8\times(-\frac{1}{2}) = 4\).

Step3: Calculate the \(y\) - coordinate

We know that \(\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}\). Using the formula \(y = r\sin\theta\), substitute \(r=-8\) and \(\sin\theta=\frac{\sqrt{3}}{2}\). Then \(y=-8\times\sin\frac{2\pi}{3}=-8\times\frac{\sqrt{3}}{2}=-4\sqrt{3}\).

Answer:

\((4,-4\sqrt{3})\)