Given the original Polar Coordinate (7,π/7) in the form (r,θ) write all the alternative forms.

Question

1. (-r, θ)
2. (r, -θ)
3. (-r, -θ)

Answer 1

Hi Vi!

This question is a little misleading, so I understand your confusion. You cannot simply negate r=7 to get the answer for question 1, that's just a different point. Instead, the question wants to know a point in polar coordinates that is the same as the original, but has the indicated signs of r and theta.

For question 1, we can start by making r negative. -7 is really our only option, because any other negative value would either be too large or too small, no mater what theta value. If you graph the point (7,pi/7) and (-7,pi/7) in a tool like desmos, you can see that these points are on opposite sides of the line theta=pi/7, meaning we need to rotate (-7,pi/7) by 180 degrees or pi radians. We can do this by simply adding pi to pi/7. So the answer to 1 is (-7,8pi/7).

Question 3 is really similar to question 2, so I'll go there next. The process is exactly the same except for how we shifted the theta value by adding pi. We could have actually subtracted pi, and that would have resulted in the same 180 degree rotation, just the other way around. This would result in the point (-7,-6pi/7), and because this has a negative theta value, it's our final answer.

For question 2, we don't have to change the value for r, because it's already positive. However, we do need to make theta negative. Because there is no other change we've made, we need to rotate the point around the origin 360 degrees, or 2pi radians. Now, when we subtract 2pi, we get (7,-13pi/7). Now, with a negative theta value, this is the final answer.

In short, these answers can come from playing around in certain graphing tools, and really getting familiar with how polar coordinates work. This question is also worded in a misleading manner, so this may be a future hurdle in this class, but it does not necessarily reflect your understanding or lack there of!

Hope this helps!