Question

Which polar coordinates below also represent that point? Check all that apply.

a. (2, startfraction 7 pi over 3 endfraction)
b. (2, startfraction 4 pi over 3 endfraction)
c. (2, negative startfraction 5 pi over 3 endfraction)
d. (negative 2, startfraction 4 pi over 3 endfraction)
e. (negative 2, negative startfraction pi over 3 endfraction)

Answer 1

The polar coordinates that represent the same point are:

a. (2, startfraction 7 pi over 3 endfraction)

b. (2, startfraction 4 pi over 3 endfraction)

c. (2, negative startfraction 5 pi over 3 endfraction)

e. (negative 2, negative startfraction pi over 3 endfraction)

Step-by-step explanation:

In polar coordinates, the radius and angle determine a point's position. Points with the same radius but differing angles will fall on the same line from the origin but at distinct angles. Options a, b, and c share the same radius of 2 units but with angles at 7π/3, 4π/3, and -5π/3, respectively. These angles are coterminal with each other as they correspond to the same terminal side when plotted, hence representing the same point. Additionally, option e features a radius of 2 units with an angle of -π/3, which also places it on the same line but in the opposite direction due to the negative radius.

However, option d has a different radius (-2), indicating a point on the same line as the others but positioned at a distance of 2 units in the opposite direction from the origin. Therefore, it doesn't represent the same point as the others, differing in both radius and direction.