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.
