Question Video: Graphing Polar Coordinates

Video Transcript

Consider the points plotted on the graph. Write down the polar coordinates of 𝐶, giving the angle 𝜃 in the range 𝜃 is greater than negative 𝜋 and less than or equal to 𝜋.

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

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