Please help, polar coordinates

Polar coordiantes can be represented as (r, θ) where r equals the radius and θ equals the angle in degrees or in radians.  To convert the cartesian coordinate (2,-2) to polar coordinate, first figure out what r is. To find out what r is, use the formula known as pythagoras theorem: ${r}^{2}={x}^{2}+{y}^{2}$ where x is the x-coordinate and y is the y-coordinate.

${r}^{2}={x}^{2}+{y}^{2}$

${r}^{2}={2}^{2}+{(-2)}^{2}$

${r}^{2}=4+{(-2)}^{2}$

${r}^{2}=4+4$

${r}^{2}=8$

$\sqrt{{r}^{2}}=\sqrt{8}$

$r=\sqrt{8}$

$r=2\sqrt{2}$

Now figure out what θ is.  To figure out what θ is, use the formula known as tangent function:

$tan(\Theta)=\frac{x}{y}$.

$tan(\Theta)=\frac{x}{y}$

$tan(\Theta)=\frac{2}{-2}$

$tan(\Theta)=-\frac{2}{2}$

$tan(\Theta)=-1$

${tan}^{-1}(tan(\Theta))={tan}^{-1}(-1)$

$\Theta ={tan}^{-1}(-1)$

$\Theta =-45°$or $\Theta =-\frac{\pi}{4}$

Because the question asks to be within the 0° ≤ θ < 360° parameter, ignore the radian answer above.  Since the degree answer is not within the 0° ≤ θ < 360° parameter, you need to change the answer to an equilivent answer that fits the 0° ≤ θ < 360°.  To do that add 360° to the degree answer.

$\Theta=-45°+360°$

$\Theta=315°$

Now put r and θ in polar cordinate form.

$(r,\Theta)$

$(2\sqrt{2}, 315°)$

To find another coordinate in polar form that is the same as the polar coordinate above that fits the 0° ≤ θ < 360°, first subtract 180° from 315°.

$\Theta=315°-180°$

$\Theta=135°$

Second, change $2\sqrt{2}$ to $-2\sqrt{2}$.

Now put r and θ in polar cordinate form.

$(r,\Theta)$

$(-2\sqrt{2},135°)$

The following image should help:

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gibsonj338's other result requires a negative radial distance.  Difficult to visualise what this means!!

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