Video: Graphing Polar Coordinates

Consider the points plotted on the graph. Write down the polar coordinates of 𝐢, giving the angle πœƒ in the range βˆ’πœ‹ < πœƒ ≀ πœ‹.

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Video Transcript

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

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

Now, π‘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 π‘Ÿ must be equal to one. But what about the angle πœƒ? We know that a full turn is two πœ‹ radians. And half a turn is πœ‹ radians. This half a turn is split into 12 subintervals. So each subinterval must represent πœ‹ by 12 radians. Our half line travels three of these subintervals. That’s three lots of πœ‹ by 12, which is πœ‹ by four. But we’re travelling in a clockwise direction. So our value of πœƒ for the polar coordinates of 𝐢 is negative πœ‹ by four. And the polar coordinates of 𝐢 are therefore one, negative πœ‹ by four. Notice that had we travelled in a counterclockwise direction, we’d have, of course, an angle of seven, πœ‹ by four. But that’s outside of the range of πœƒ given.

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