Labeling Negative Polar Coordinates

Question

i have a quiz coming up tomorow and i have this major question.. its a question about how to plot it exactly when it is negative. Let me go through the whole set:

1) ($4$ , $60^\circ$)

2)($-4$ , $270^\circ$)

3)($-2$ , $-135^\circ$)

4)($-8$ , $-270^\circ$)

5)($6$ , $-45^\circ$)

And here is the diagram of polar coordinates..

So i can perfectly do #1 and #6, you just find the angle you drawing and then go the amount in R(for example 4). But for the negative and double negative im troubled.. i already have anwser so i dont need that at all.. i just owant to know the logic on how to plot polar coordinates for negative and double negatives.. Thanks a lot!

Answer 1

An argument of 45 degrees means going from the positive $x$ axis 45 degrees in the counter-clockwise direction. So an argument of -45 degrees means going 45 degrees clockwise from the positive $x$ axis.

As for the modulus, by convention $(-1,\theta)$ is the same as $(1,-\theta)$.

Answer 2

A ordered pair with a negative distance component is the same thing as one with the opposite angle component. (plus 180 degreees) Negative angle components are the same thing as themselves plus 360 degrees.

For example:

$$(-4, 90)$$ $$(4, 270)$$ $$$$ $$(5, -30)$$ $$(5, 330)$$ $$$$ $$(-10, -60)$$ $$(-10, 300)$$ $$(10, 480)$$ $$(10, 20)$$

Answer 3

Think about the number line being rotated. For $\theta=0$, you get the standard number line where $r>0$ on the right of the origin and $r<0$ on the left. So for example the point $(-4,0)$ is exactly the same location in polar coordinates as it is in rectangular coordinates.

For other angles, the positive values for $r$ go toward where the angle is labeled on any circle, starting at the positive $x-$axis and going counterclockwise. So for example $(-4,60^{\circ})$ would be in the third quadrant because if you rotate your number line so that the positive $r$ values head toward where the 60 degree angle is, then the negative ones are....

In the case where both $r$ and $\theta$ are negative, first rotate the number line clockwise and then you will know which direction the positive $r$ values head toward. So for example $(-2,-90^{\circ})$ would be up 2 units from the origin...

Hope this helps!

Answer 4

$(-4,270) \rightarrow (4,90). $ So a change of sign of $r$ means add or subtract 180 degrees for polar angle in the $ opposite $ quadrant.

Answer 5

$\newcommand{\degree}{^\circ}$ A negative radial coordinate is rotated half-circle from a positive radial coordinate, that is: $$(-r, \theta) \equiv (r, \theta\pm 180\degree)$$ [NB: either add or subtract $180\degree$ once; also see below if the angle is still outside the $0\degree$ to $360\degree$ range.]

$$(-5, 45\degree) \equiv (5, 125\degree) \\ (-10, 190\degree) \equiv (10, 10\degree)$$

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A negative angle coordinate is clockwise from the x-axis rather than widdershins, and can be made positive with sufficient full-circle rotations (usually one).

$$(r, -\theta) \equiv (r, n360\degree-\theta) \qquad [\forall n\in\Bbb N]$$

$$(15, -7\degree) \equiv (15, 353\degree)\\(26, -279\degree ) \equiv (26, 81\degree)\\ (2, -596\degree) \equiv (2, 124\degree)$$