Question

convert the polar coordinate \\(\left(8, \frac{3\pi}{4}\
ight)\\) to cartesian coordinates. enter exact values. \\(x = \\) \\(y = \\) question help: worked example 1

Explanation:

Step1: Recall the conversion formulas

For polar coordinates \((r,\theta)\) to Cartesian coordinates \((x,y)\), we use \(x = r\cos\theta\) and \(y=r\sin\theta\). Here \(r = 8\) and \(\theta=\frac{3\pi}{4}\).

Step2: Calculate the \(x\) - coordinate

We know that \(\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}\). Substitute \(r = 8\) and \(\cos\theta\) into \(x=r\cos\theta\):
\(x=8\times\cos\frac{3\pi}{4}=8\times(-\frac{\sqrt{2}}{2})=- 4\sqrt{2}\)

Step3: Calculate the \(y\) - coordinate

We know that \(\sin\frac{3\pi}{4}=\frac{\sqrt{2}}{2}\). Substitute \(r = 8\) and \(\sin\theta\) into \(y = r\sin\theta\):
\(y=8\times\sin\frac{3\pi}{4}=8\times\frac{\sqrt{2}}{2}=4\sqrt{2}\)

Answer:

\(x=-4\sqrt{2}\)
\(y = 4\sqrt{2}\)