Video: Multiple Representations of Polar Coordinates

Which of the ordered pairs (4, −30°), (4, 330°), (4, 390°), and (4, −390°) does not describe the position of Point 𝐵 in the diagram?

04:03

Video Transcript

Which of the ordered pairs four, negative 30 degrees; four, 330 degrees; four, 390 degrees; and four, negative 390 degrees does not describe the position of point 𝐵 in the diagram?

In this question, the term ordered pairs refers to polar coordinates. 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. From the diagram, we can see that the distance from the origin of point 𝐵 is four units. And the angle between the polar axis and point 𝐵 is negative 30 degrees, i.e., 30 degrees in the clockwise direction. Hence, the ordered pair four, negative 30 degrees given to us in the question does describe the position of point 𝐵.

We can also measure the angle between the polar axis and point 𝐵 in the counterclockwise direction. Since each quadrant bounds an angle of 90 degrees, measuring three-quarters of a full rotation in the counterclockwise direction from the polar axis is equivalent to measuring 270 degrees in the counterclockwise direction from the polar axis. Hence, the angle between the polar axis and the point 𝐵 in the counterclockwise direction is 270 degrees plus 𝑎, where 𝑎 is the angle in the fourth quadrant labelled in the diagram.

We compute 𝑎 to be 90 degrees minus 30 degrees, which equals 60 degrees. Hence, the angle between the polar axis and point 𝐵 in the counterclockwise direction is 270 degrees plus 60 degrees, which equals 330 degrees. So the position of point 𝐵 can be described by the ordered pair four, 330 degrees.

Now, note that one possibility with a set of ordered pairs that describe the position of point 𝐵 is 𝑟, 𝜃, where 𝑟 is equal to four and 𝜃 is equal to negative 30 minus any integer multiple of 360. This is because we can add a number of full rotations of 360 degrees in the clockwise direction to the original negative 30-degree angle between the polar axis and point 𝐵. Letting 𝑛 equal one in the formula stated on the screen, we obtain that the position of point 𝐵 can also be described by the ordered pair four, negative 30 minus 360 degrees, i.e., four, negative 390 degrees.

So out of the options given to us in the question, the only option remaining for the ordered pair that does not describe the position of point 𝐵 is four, 390 degrees. Since 390 equals 360 plus 30, the point four, 390 degrees represents the point which makes an angle of 30 degrees in the counterclockwise direction with the polar axis and lies at a distance of four units from the origin. Let’s call this point 𝑝.

Now, it is clear that point 𝑝 does not describe the position of point 𝐵 as point 𝑝 lies above the polar axis at an angle of 30 degrees in the counterclockwise direction. And point 𝐵 lies below the polar axis at an angle of 30 degrees in the clockwise direction.

So the answer to the question is the ordered pair four, 390 degrees.

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