convert the following point from polar to cartesian coordinates. write the exact answer as an ordered pair…

Explanation:

Step1: Recall the conversion formulas

The conversion formulas from polar coordinates ((r,\theta)) to Cartesian coordinates ((x,y)) are (x = r\cos\theta) and (y=r\sin\theta). Here (r=- 8) and (\theta=\frac{5\pi}{6}).

Step2: Calculate the (x) - coordinate

We know that (\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}). Substitute (r = - 8) and (\theta=\frac{5\pi}{6}) into (x = r\cos\theta): [ \begin{align*} x&=-8\times\cos\frac{5\pi}{6}\ &=-8\times\left(-\frac{\sqrt{3}}{2}\right)\ &=4\sqrt{3} \end{align*} ]

Step3: Calculate the (y) - coordinate

We know that (\sin\frac{5\pi}{6}=\frac{1}{2}). Substitute (r = - 8) and (\theta=\frac{5\pi}{6}) into (y = r\sin\theta): [ \begin{align*} y&=-8\times\sin\frac{5\pi}{6}\ &=-8\times\frac{1}{2}\ &=-4 \end{align*} ]

Answer:

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