Question

For questions 1-5, the polar coordinates of a point are given. Find the rectangular coordinates of each point.

1. (5,(π / 4))

2. (-2,(π / 6))

3. (-1,(-2π / 3))

4. (3,225°)

5. (-3,330°)

Answer 1

QUESTION 1

We want to convert
`(5,(\pi)/(4))` from polar coordinates to rectangular coordinates.

We use the formula;

`x=r\cos \theta`

`y=r\sin \theta`

We plug in
`r=5` and
`\theta=(\pi)/(4)` to obtain;

`x=5\cos (\pi)/(4)=(5√(2))/(2)`

`y=5\sin (\pi)/(4)=(5√(2))/(2)`

The rectangular coordinate is
`((5√(2))/(2),(5√(2))/(2))`

QUESTION 2

We want to convert
`(-2,(\pi)/(6))` from polar coordinates to rectangular coordinates.

We use the formula;

`x=r\cos \theta`

`y=r\sin \theta`

We plug in
`r=-2` and
`\theta=(\pi)/(6)` to obtain;

`x=-2\cos (\pi)/(6)=-1`

`y=-2\sin (\pi)/(6)=-√(3)`

The rectangular coordinate is
`(-1,-√(3))`

QUESTION 3

We want to convert
`(-1,(-2\pi)/(3))` from polar coordinates to rectangular coordinates.

We use the formula;

`x=r\cos \theta`

`y=r\sin \theta`

We plug in
`r=-1` and
`\theta=(-2\pi)/(3)` to obtain;

`x=-1\cos (-2\pi)/(3)=(1)/(2)`

`y=-1\sin (-2\pi)/(3)=(√(3))/(2)`

The rectangular coordinate is
`((1)/(2),(√(3))/(2))`

QUESTION 4

We want to convert
`(3,225\degree)` from polar coordinates to rectangular coordinates.

We use the formula;

`x=r\cos \theta`

`y=r\sin \theta`

We plug in
`r=3` and
`\theta=225\degree` to obtain;

`x=3\cos (225\degree)=-3(√(2))/(2)`

`y=3\sin(225\degree)=-(3√(2))/(2)`

The rectangular coordinate is
`(-(3√(2))/(2),-3(√(2))/(2))`

QUESTION 5

We want to convert
`(-3,330\degree)` from polar coordinates to rectangular coordinates.

We use the formula;

`x=r\cos \theta`

`y=r\sin \theta`

We plug in
`r=-3` and
`\theta=330\degree` to obtain;

`x=-3\cos (330\degree)=-(3√(3))/(2)`

`y=-3\sin(330\degree)=(3)/(2)`

The rectangular coordinate is
`(-(3√(3))/(2),(3)/(2))`