Question

The polar coordinates of a point are 3π 4 and 7.00 m. What are its Cartesian coordinates (in m)? (x, y) = 6.06,−3.5 m

Answer 1

a)
`\left(x,y\right)=\left(4.95,-4.95\right)`

b)
`r\angle\theta = 7\angle0.5236\,\text{radians}`

Explanation:

Polar coordinates are represented as:
`r\angle\theta`, where 'r' is the length (or magnitude) of the line, and '
`\theta`' is the angle measured from the positive x-axis.

in our case:

`7\angle(3\pi)/(4)`

to covert the polar to cartesian:

`x = rcos(\theta)`

`y = rsin(\theta)`

we can plug in our values:

`x = 7\cos{(3\pi)/(4)} = -7(√(2))/(2)`

`y = 7\sin{(3\pi)/(4)} = 7(√(2))/(2)`

the Cartesian coordinates are:

`\left(x,y\right)=\left(-7(√(2))/(2),7(√(2))/(2)\right)`

`\left(x,y\right)=\left(4.95,-4.95\right)`

(b) to convert (x,y) = (6.06,-3.5)

we'll use the pythagoras theorem to find 'r'

`r^2 = x^2+y^2`

`r^2 = (6.06)^2+(-3.5)^2`

`r = √(48.97) \approx 7`

the angle can be found by:

`tan(\theta) = (y)/(x)`

`tan(\theta) = (3.5)/(6.06)`

`\theta = \arctan{left((3.5)/(6.06)\right)}`

`\theta = 0.5236 \text{radians}`

to convert radians to degrees:

`\theta = 0.5236 * (180)/(\pi) \approx 30^\circ`

writing in polar coordinates:

`r\angle\theta = 7\angle30^\circ\,\,\text{OR}\,\,7\angle0.5236\,\text{radians}`