Question

Which of the following polar coordinates represents the same location as (4,200)?

A) 4,−200
B) 4,−160
C) 4,−110
D) 4,70
E) 4,160

Answer 1

Option (A) (4,-200) represents the same location as (4,200). By adding 360 degrees to -200, we get 160, which is a full rotation from 200, thus indicating the same point.

Step-by-step explanation:

The given polar coordinates of the student's question are (4,200). To find which of the choices represents the same location in polar coordinates, we must understand that adding or subtracting full rotations (360 degrees) will result in the same location. Moreover, polar coordinates are periodic with a period of 360 degrees, so we can add or subtract multiples of 360 degrees to find equivalent coordinates. Given this, equivalent polar coordinates can also be found by simply changing the sign of the angle and adjusting it into the conventional range of -180 degrees to 180 degrees.

Let's analyze the given options:

• (4,−200) represents the same point as (4,200) because if we add 360 degrees to -200, we get 160, which is a full rotation from 200.
• (4,−160) is not equivalent because it represents a different position.
• (4,−110) is not equivalent because it represents a different position.
• (4,70) is not equivalent because it does not account for a full rotation difference from 200 degrees.
• (4,160) is not equivalent, because 200 - 160 = 40, which does not represent a full rotation.

Therefore, the options that represent the same location as (4,200) are (4,−200) because by adding 360 degrees, it results in the angle of 160 degrees, which is a complete circle away from 200 degrees, signifying the same point.