Question

WILL AWARD MEDAL! NEED HELP ASAP :)
Determine two pairs of polar coordinates for the point (2, -2) with 0° ≤ θ < 360°.

(2 square root of 2, 225°), (-2 square root of 2, 45°)
(2 square root of 2, 135°), (-2 square root of 2, 315°)
(2 square root of 2, 315°), (-2 square root of 2, 135°)
(2 square root of 2, 45°), (-2 square root of 2, 225°)

Answer 1

Well, Say we have a circle around the origin with the point (2, -2) on it:

We'd like to know two things. The radius of this circle and the angle where this point is on it.
First let's calculate the radius. We know the point (2, -2) is on the circle, and that the center of the circle is the origin (0,0) so the distance between those two points must be the radius:
$$
r = \sqrt{(2 - 0)^2 + (-2 -0)^2} = \sqrt{4 + 4} = \sqrt{4 \cdot 2} = 2 \sqrt{2}
$$
Now let's see what angle that point is on the circle. To do so we can use the ratio between the x, y components of the point to tell us the tangent value for the wanted angle:
$$
tan(\alpha) = \frac{-2}{2} = -1
$$Now we can find one of the angles using the inverse tangent function:
$$
\alpha = arctan(-1) = -45^{\circ} = 360^\circ - 45^\circ = 315^\circ
$$However this is not the only solution as the point (-2, 2) which is exactly on the other side of the circle compared to (2, -2) would yield the same ratio and therefore the same tangent.
Since this point is on the other side of the circle we could simply add or subtract \(180^\circ\) to the angle we have to find it:
$$
\beta = \alpha - 180^\circ = 315^\circ - 180 ^\circ = 135^\circ
$$
Now to construct our coordinates we have to take the angles and determine whether our point is in the direction of the angle or the reverse direction.
We can tell that the angle \(315^\circ\) is in the fourth quadrant, where x is positive and y is negative, which is where our point (2, -2) is located.
So therefore it is 'in the direction' of \(315^\circ\) and in the reverse direction of \(135^\circ\), so now we can say:
\(
P_1 = (\alpha, r) = (315^\circ, 2\sqrt{2}) \\
P_2 = (\beta, -r) = (135^\circ, -2\sqrt{2}) \\
\)

Answer 2

@pitamar THANK YOU! Would u look at one more?
Find the rectangular coordinates of the point with the polar coordinates (1, 1 divided by 2 pi).

(0, -1)
(1, 0)
(0, 1)
(-1, 0)

Answer 3

Ok, well in case your coordinates are:
$$
P = \Big(1, \frac{1}{2} \pi \Big) = \Big(1, \frac{\pi}{2}\Big)
$$ Then we can say:
\(\frac{\pi}{2}\) is basically \(90^\circ\). If we'd make a circle with radius of 1 (because our polar coordinate tells us so) and wanted to check the coordinates at a given angle, say \(90^\circ\) we could do:
$$
x = \cos(\alpha) \cdot r \\
y = \sin(\alpha) \cdot r \\
\;\\
\alpha = \frac{\pi}{2} \qquad r = 1 \\
x = \cos(\frac{\pi}{2}) \cdot 1 = 0 \cdot 1 = 0 \\
y = \sin(\frac{\pi}{2}) \cdot 1 = 1 \cdot 1 = 1
$$ So are coordinates are (0, 1)