Determine if the given polar coordinates represent the same point.
-$$( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )$$

Question

Quizplus · Accepted Answer

To determine if two polar coordinates represent the same point, we need to convert them to rectangular coordinates and compare their values.

For the first set of polar coordinates, we have: 
$r = 5$
$	heta = \pi / 3$
Converting to rectangular coordinates: 
$x = r cos 	heta = 5 cos (\pi / 3) = 5 (1/2) = 2.5$
$y = r sin 	heta = 5 sin (\pi / 3) = 5 (\sqrt{3} / 2) = 2.5 \sqrt{3}$
Therefore, the first set of polar coordinates represents the point $(2.5, 2.5 \sqrt{3})$

For the second set of polar coordinates, we have: 
$r = -5$
$	heta = -2 \pi / 3$
Converting to rectangular coordinates: 
$x = r cos 	heta = -5 cos (-2 \pi / 3) = -5 (-1/2) = 2.5$
$y = r sin 	heta = -5 sin (-2 \pi / 3) = -5 (\sqrt{3}/2) = 2.5 \sqrt{3}$
Therefore, the second set of polar coordinates also represents the point $(2.5, 2.5 \sqrt{3})$

Since both sets of polar coordinates represent the same point in rectangular coordinates, the answer is true and the coordinates represent the same point.

Determine If the Given Polar Coordinates Represent the Same Point (5,π/3),(−5,−2π/3)( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )(5,π/3),(−5,−2π/3)

Determine If the Given Polar Coordinates Represent the Same Point $( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )$