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

Explanation:

Step1: Recall the conversion formulas

For a polar point ((r,\theta)), the (x)-coordinate is (x = r\cos\theta) and the (y)-coordinate is (y=r\sin\theta). Here (r = 5) and (\theta=\frac{2\pi}{3}).

Step2: Calculate the (x)-coordinate

We know that (\cos\frac{2\pi}{3}=\cos(\pi-\frac{\pi}{3})=-\cos\frac{\pi}{3}). Since (\cos\frac{\pi}{3}=\frac{1}{2}), then (x = 5\cos\frac{2\pi}{3}=5\times(-\frac{1}{2})=-\frac{5}{2}).

Step3: Calculate the (y)-coordinate

We know that (\sin\frac{2\pi}{3}=\sin(\pi - \frac{\pi}{3})=\sin\frac{\pi}{3}). Since (\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}), then (y = 5\sin\frac{2\pi}{3}=5\times\frac{\sqrt{3}}{2}=\frac{5\sqrt{3}}{2}).

Answer:

(\left(-\frac{5}{2},\frac{5\sqrt{3}}{2}\right))