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.
