Question

Convert the point to rectangular coordinates. Show all work and give exact values (no decimals).

Answer 1

Given:

`(-4,(7\pi)/(6))`
`\text{Here r=-4 and }\theta=(7\pi)/(6)\text{.}`

Recall the formula for rectangular coordinates

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

`\text{Substitute r=-4 and }\theta=(7\pi)/(6)\text{ in x=rcos}\theta,\text{ we get}`

`x=-4\cos ((7\pi)/(6))`

`Use\cos ((7\pi)/(6))=\cos (\pi+(\pi)/(6))=-\cos ((\pi)/(6))=-\frac{\sqrt[]{3}}{2}\text{.}`

`x=-4(\frac{-\sqrt[]{3}}{2})=2\sqrt[]{3}_{}`

`\text{Substitute r=-4 and }\theta=(7\pi)/(6)\text{ in y=rsin}\theta,\text{ we get}`

`y=-4\sin ((7\pi)/(6))`

`Use\text{ sin}((7\pi)/(6))=sin(\pi+(\pi)/(6))=-sin((\pi)/(6))=-(1)/(2)\text{.}`

`y=-4(-(1)/(2))=2`

Hence the rectangular coordinates are

`(x,y)=(2\sqrt[]{3},\text{ 2)}`