Question

Determine two pairs of polar coordinates for the point (5, -5) with 0° ≤ θ < 360°.

Answer 1

(5,-5) can be represented in polar form by

`(2√(5),315^\circ)` and
`(-2√(5),135^\circ)`

Explanation:

polar coordinates use a distance and an angle

it would be like (x,y) but x is distance from origin to point and y is the angle measured counterclockwise from the positive x-axis.

for (5,-5)

first find the distance to that point using distance formula

distance from (0,0) to (5,-5) is

`D=√((0-5)^2+(0-(-5))^2)`

`D=√(25+25)`

`D=5√(2)`

so our point has to be in the form
`(x,y)` where
`\mid x\mid=5√(2)`

now finding the degree

using inverse tangent

`tan^(-1)((-5)/(5))=-45^\circ`

if we look on the graph, it is also 360-45=315 degrees from positive x axis

so one polar coordiante is
`(2√(5),315^\circ)`

the other one is in the oposite side

we add or subtract 180 degrees and make the sign of x negative to go in the oposite direction

subtraction 180 to get 135 degrees

so the other point is
`(-2√(5),135^\circ)`

(5,-5) can be represented in polar form by

`(2√(5),315^\circ)` and
`(-2√(5),135^\circ)`

Answer 2

Explanation:

Alright, lets get started.

The given co-ordinates are (5,-5)

This is the form of (x,y).

We need to find this co-ordinate into polar form means in (r,Θ) form.

The formula for r is :

`r=√(x^2+y^2)`

`r=\sqrt{5^2+(-5)^(2) }`

`r=√(25+25)`

`r=√(50)`

`r=5√(2)`

Now the Θ part

The formula for Θ is : Θ =
`tan^(-1)((y)/(x) )`

Θ=
`tan^(-1)((-5)/(5))`

Θ=
`tan^(-1)(-1)`

There are two angles within the intervals where tan is -1. That angles are 135 and 315.

So two polar co-ordinates are :
`(5√(2), 135)` and
`(5√(2),315)` : Answer

Hope it will help :)