The following polar coordinates are multiple representations of the same point: (6, 5pi/4) and (6, 13pi/4)?

Question

Is this answer true or false?
How do I set this up and figure out the answer?

Answer 1

True since r = constant and the sin 5pi/4 = sin 13pi/4 since both thetas are even multiples of 2pi and sin (4pi/4 + pi/4) = sin (12pi/4 + pi/4)

sin (4pi/4 + pi/4) = sin 4pi/4 cos pi/4 + cos 4pi/4 sin pi/4

since sin 4pi/4 = 0 we have left 1 * sin pi/4 = 1/2

similarly

sin (12pi/4 + pi/4) = sin 3pi cos pi/4 + cos 3pi sin pi/4

since sin 3pi = 0 we have left 1* sin pi/4 = 1/2

Answer 2

First, note that the two coordinates have the same radii. Now you can proceed to check their phase ( angles).

Polar coordinates that have same radii do not change after a phase(angle) addition or subtraction by 2pi.

So, just add 2pi to the lower angle ( 5pi/4 ) or better, subtract 2pi from the larger one(13pi/4) , until it becomes lower than or equal to the lower angle.

Here.. it becomes equal

13pi/4 - 2*pi = (13-8) * pi / 4 = 5pi/4

If it becomes lower than the lowest angle in the problem, then they shouldn't be equal.

For example, if we consider 4pi/5 and 23pi/5 ... The greater angle is 23pi/5. So subtracting 2pi's from 23pi/5 , we have ..

23pi/5 - 2pi = 13pi/5 - 2*pi = 3pi/5 , which is lesser than 4pi/5.

Hence in the example I considered above, the angles are not equal.

Love and Light.

Answer 3

In Polar coordinates, the angle 2pi is the same as the angle 0, and likewise 2pi plus x is the same as x.

So since 13pi/4 > 2pi then you can subract 2pi from that angle until you get something between zero and 2pi. Then you can compare with other angles and decide if they are the same.

Answer 4

True.

In polar coordinates, two points are the same if and only if:

Their r values are equal
Θ1 = Θ2 ± 2kπ where k is an integer.

their r values are both 6,
13π/4 = 5π/4 + 8π/4
In this case, Θ2 = Θ1 + 2(1)π