Question

convert the polar coordinate (left(8, \frac{5pi}{3}
ight)) to cartesian coordinates.
(x =)
(y =)

Explanation:

Step1: Recall the conversion formulas

For polar coordinates \((r,\theta)\) to Cartesian coordinates \((x,y)\), the formulas are \(x = r\cos\theta\) and \(y=r\sin\theta\). Here \(r = 8\) and \(\theta=\frac{5\pi}{3}\).

Step2: Calculate the value of \(x\)

We know that \(\cos\frac{5\pi}{3}=\cos(2\pi-\frac{\pi}{3})=\cos\frac{\pi}{3}\) (using the identity \(\cos(2\pi - \alpha)=\cos\alpha\)). Since \(\cos\frac{\pi}{3}=\frac{1}{2}\), then \(x = 8\times\cos\frac{5\pi}{3}=8\times\frac{1}{2}=4\).

Step3: Calculate the value of \(y\)

We know that \(\sin\frac{5\pi}{3}=\sin(2\pi-\frac{\pi}{3})=-\sin\frac{\pi}{3}\) (using the identity \(\sin(2\pi - \alpha)=-\sin\alpha\)). Since \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), then \(y = 8\times\sin\frac{5\pi}{3}=8\times(-\frac{\sqrt{3}}{2})=- 4\sqrt{3}\).

Answer:

\(x = 4\)
\(y=-4\sqrt{3}\)