### Video Transcript

The polar coordinates of point πΈ are two, 80 degrees. Which of the points πΉ: two, 380 degrees; πΊ: two, 440 degrees; π»: two, negative 80 degrees; or πΌ: four, 160 degrees is coincident with point πΈ?

Recall that we say the polar coordinates of a point π are the ordered pair π, π if π is the angle between the polar axis and the line ππ and π is the distance from π to π . Remember that the letter π denotes the origin. By convention, the angle π is measured in the counterclockwise direction from the polar axis if it is positive and in the clockwise direction if it is negative.

Letβs plot the point πΈ using its polar coordinates. The point πΈ lies 80 degrees in the counterclockwise direction from the polar axis, at a distance of two units from the origin. We want to find which of the points πΉ, πΊ, π», or πΌ given to us in the question is coincident with point πΈ. Recall that two points are called coincident if they are actually the same point, but just written in different ways.

Therefore, since a complete rotation is given by 360 degrees, the point πΈ, represented by the polar coordinates two, 80 degrees, is coincident with all points from the form π, π. Where π is equal to two and π is equal to 80 plus any integer multiple of 360. If we let π equal one, then the point πΈ: two, 80 degrees is coincident with the point two, 80 plus 360 degrees, which simplifies to two, 440 degrees. This corresponds to the point πΊ given to us in the question. So it seems like the point πΊ: two, 440 degrees is our final answer.

Letβs confirm that the remaining points πΉ, π», and πΌ are not coincident with the point πΈ. Letβs have a look at the point πΌ: four, 160 degrees. It is clear that the point πΌ is not coincident with the point πΈ as the distance of πΌ from the origin is four, which is not equal to two, the distance of πΈ from the origin.

Letβs have a look at the point πΉ: two, 380 degrees. We have that 380 is equal to 360 plus 20. Hence, measuring 380 degrees in the counterclockwise direction from the polar axis is the same as measuring 20 degrees in the counterclockwise direction from the polar axis. The point πΉ lies at a distance of two units from the origin.

It is now clear that even though the points πΈ and πΉ lie the same distance away from the origin, they are not coincident. As the angle between the polar axis and the point πΉ is 20 degrees in the counterclockwise direction, which is not equal to 80 degrees, the angle between the polar axis and the point πΈ in the counterclockwise direction.

Finally, letβs have a look at the point π»: two, negative 80 degrees. The point π» lies 80 degrees in the clockwise direction from the polar axis, at a distance of two units from the origin. It is now clear that the point π» is not coincident with the point πΈ. As the point π» lies below the polar axis at an angle of 80 degrees in the clockwise direction. And the point πΈ lies above the polar axis at an angle of 80 degrees in the counterclockwise direction. So the only option remaining is the point πΊ: two, 440 degrees. And we have seen previously that the point πΊ is in fact coincident with the point πΈ.