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,(\pi)/(6)) \\ r=2,\theta=(\pi)/(6) \end{gathered}`

Substitute them in the form above

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

b) The second point is

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

c) The third point is

`(4.4,(13\pi)/(12))`
`\begin{gathered} x=4.4\cos ((13\pi)/(12))=4.4(-\frac{\sqrt[]{6}+\sqrt[]{2}}{4})=(-1.1\sqrt[]{6}-1.1\sqrt[]{2}) \\ y=4.4\sin ((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,(5\pi)/(3))`
`\begin{gathered} x=4.4\cos ((5\pi)/(3))=4.4((1)/(2))=2.2 \\ y=4.4\sin ((5\pi)/(3))=4.4(-\frac{\sqrt[]{3}}{2})=-2.2\sqrt[]{3} \\ (x,y)=(2.2,-2.2\sqrt[]{3)} \end{gathered}`