Question Video: Graphing Polar Coordinates Mathematics • 12th Grade

Consider the points plotted on the graph. Write down the polar coordinates of ๐ถ, giving the angle ๐œƒ in the range โˆ’๐œ‹ < ๐œƒ โ‰ค ๐œ‹.

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Video Transcript

Consider the points plotted on the graph. Write down the polar coordinates of ๐ถ, giving the angle ๐œƒ in the range ๐œƒ is greater than negative ๐œ‹ and less than or equal to ๐œ‹.

Weโ€™re interested in the point ๐ถ. And we want to know its polar coordinates. Remember, these are of the form ๐‘Ÿ, ๐œƒ. Letโ€™s add a half line or a ray from the pole to point ๐ถ. Our job is going to be to work out the value of ๐‘Ÿ, thatโ€™s the length of our half line, and ๐œƒ, the angle that this half line makes with the positive ๐‘ฅ-axis. And since weโ€™re told that ๐œƒ must be greater than negative ๐œ‹ and less than or equal to ๐œ‹, weโ€™re going to travel in a clockwise direction.

Now, ๐‘Ÿ is quite easy to calculate. We follow the grid around. And we see that the point is located exactly one unit from the pole. So ๐‘Ÿ must be equal to one. But what about the angle ๐œƒ? We know that a full turn is two ๐œ‹ radians. And half a turn is ๐œ‹ radians. This half a turn is split into 12 subintervals. So each subinterval must represent ๐œ‹ by 12 radians. Our half line travels three of these subintervals. Thatโ€™s three lots of ๐œ‹ by 12, which is ๐œ‹ by four. But weโ€™re travelling in a clockwise direction. So our value of ๐œƒ for the polar coordinates of ๐ถ is negative ๐œ‹ by four. And the polar coordinates of ๐ถ are therefore one, negative ๐œ‹ by four. Notice that had we travelled in a counterclockwise direction, weโ€™d have, of course, an angle of seven, ๐œ‹ by four. But thatโ€™s outside of the range of ๐œƒ given.

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