Question

In the diagram below, assume that all points are given in polar coordinates. Determine the rectangular coordinates for each point. Visually check your answers to ensure they make sense.(r,θ)=(2,π/6) corresponds to (x,y)=(r,θ)=(2.4,π/2) corresponds to (x,y)=(r,θ)=(4.4,13π/12) corresponds to (x,y)=(r,θ)=(4.4,5π/3) corresponds to (x,y)=

Answer 1

The rectangular coordinates of the polar form are

$x=r\cos \theta$$y=r\sin \theta$

a) The first point is

$\begin{gathered} (2,\frac{\pi}{6}) \\ r=2,\theta=\frac{\pi}{6} \end{gathered}$

Substitute them in the form above

$\begin{gathered} x=2\cos (\frac{\pi}{6})=2(\frac{\sqrt[]{3}}{2})=\sqrt[]{3} \\ y=2\sin (\frac{\pi}{6})=2(\frac{1}{2})1 \\ (x,y)=(\sqrt[]{3},1) \end{gathered}$

b) The second point is

$(2.4,\frac{\pi}{2})$$\begin{gathered} x=2.4\cos (\frac{\pi}{2})=2.4(0)=0 \\ y=2.4\sin (\frac{\pi}{2})=2.4(1)=2.4 \\ (x,y)=(0,2.4) \end{gathered}$

c) The third point is

$(4.4,\frac{13\pi}{12})$$\begin{gathered} x=4.4\cos (\frac{13\pi}{12})=4.4(-\frac{\sqrt[]{6}+\sqrt[]{2}}{4})=(-1.1\sqrt[]{6}-1.1\sqrt[]{2}) \\ y=4.4\sin (\frac{13\pi}{12})=4.4(\frac{-\sqrt[]{6}+\sqrt[]{2}}{4})=(-1.1\sqrt[]{6}+1.1\sqrt[]{2}) \\ (x,y)=\lbrack(-1.1\sqrt[]{6}-1.1\sqrt[]{2}),(-1.1\sqrt[]{6}+1.1\sqrt[]{2})\rbrack \end{gathered}$

d) The fourth point is

$(4.4,\frac{5\pi}{3})$$\begin{gathered} x=4.4\cos (\frac{5\pi}{3})=4.4(\frac{1}{2})=2.2 \\ y=4.4\sin (\frac{5\pi}{3})=4.4(-\frac{\sqrt[]{3}}{2})=-2.2\sqrt[]{3} \\ (x,y)=(2.2,-2.2\sqrt[]{3)} \end{gathered}$