Question Video: Multiple Representations of Polar Coordinates Mathematics • 12th Grade

The polar coordinates of Point 𝐸 are (2, 80°). Which of the points 𝐹 (2, 380°), 𝐺 (2, 440°), 𝐻 (2, −80°), or 𝐼 (4, 160°) is coincident with Point 𝐸?

04:10

Video Transcript

The polar coordinates of point 𝐸 are two, 80 degrees. Which of the points 𝐹: two, 380 degrees; 𝐺: two, 440 degrees; 𝐻: two, negative 80 degrees; or 𝐼: four, 160 degrees is coincident with point 𝐸?

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

Let’s plot the point 𝐸 using its polar coordinates. The point 𝐸 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 𝐹, 𝐺, 𝐻, or 𝐼 given to us in the question is coincident with point 𝐸. 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 𝐸, represented by the polar coordinates two, 80 degrees, is coincident with all points from the form 𝑟, 𝜃. Where 𝑟 is equal to two and 𝜃 is equal to 80 plus any integer multiple of 360. If we let 𝑛 equal one, then the point 𝐸: two, 80 degrees is coincident with the point two, 80 plus 360 degrees, which simplifies to two, 440 degrees. This corresponds to the point 𝐺 given to us in the question. So it seems like the point 𝐺: two, 440 degrees is our final answer.

Let’s confirm that the remaining points 𝐹, 𝐻, and 𝐼 are not coincident with the point 𝐸. Let’s have a look at the point 𝐼: four, 160 degrees. It is clear that the point 𝐼 is not coincident with the point 𝐸 as the distance of 𝐼 from the origin is four, which is not equal to two, the distance of 𝐸 from the origin.

Let’s have a look at the point 𝐹: 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 𝐹 lies at a distance of two units from the origin.

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

Finally, let’s have a look at the point 𝐻: two, negative 80 degrees. The point 𝐻 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 𝐻 is not coincident with the point 𝐸. As the point 𝐻 lies below the polar axis at an angle of 80 degrees in the clockwise direction. And the point 𝐸 lies above the polar axis at an angle of 80 degrees in the counterclockwise direction. So the only option remaining is the point 𝐺: two, 440 degrees. And we have seen previously that the point 𝐺 is in fact coincident with the point 𝐸.