Consider the shown points. Which one has the polar coordinates three, 15𝜋 by four?
We’re given a diagram with five points plotted on a polar diagram: points 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸. We need to decide which of these points has the coordinates three, 15𝜋 by four. Let’s start by recalling what we mean by a polar coordinates. If the point 𝑃 has the polar coordinates 𝑟, 𝜃, where our value of 𝑟 is greater than or equal to zero, then 𝑟 represents the distance the point 𝑃 is from the origin. Alternatively, this is the size of the vector 𝐎𝐏.
Next, we know the value of 𝜃 will be the angle that the vector 𝐎𝐏 makes with the positive 𝑥-axis measured counterclockwise. For polar coordinates, this is often measured in radians. The point we’re given in the question is the point three, 15𝜋 by four. So our value of 𝑟 is three, and our value of 𝜃 is 15𝜋 by four.
We can see that this is measured in radians. There’s a few different ways we could tackle this question. For example, we might be tempted to find polar coordinate representations of all five of the points given to us in the diagram. However, instead, we’re going to use the coordinates given to us to find where they could be in our diagram.
First, we know that our value of 𝑟 is equal to three. This means the distance that our point is from the origin is equal to three. So we could ask the question which points at distance three from the origin. Of course, we know that the set of all points at distance three from the origin will trace a circle of radius three centered at the origin.
In fact, since we’re given a polar diagram, we can just sketch this onto our graph. So because this coordinate has an 𝑟-value of three and every single point of distance three from the origin lies on this circle, our point must lie on this circle. And we can see that the point 𝐶 does not lie on this circle. Therefore, the point 𝐶 can’t have polar coordinates three, 15𝜋 by four.
Now, let’s move on to our angle. We’re told that the value of 𝜃 is 15𝜋 by four. And this tells us that the angle the vector 𝐎𝐏 makes with the positive 𝑥-axis measured counterclockwise will be 15𝜋 by four. And, in fact, we’re given a polar diagram. So certain rays of different values of 𝜃 are actually given to us in the diagram.
But we can see that 15𝜋 by four is bigger than all of these values. And we know this is because one full turn around our axis will give us a value of two 𝜋. And, in fact, we can see that the value of 15𝜋 by four is bigger than two 𝜋. This means it’s bigger than one full turn. So we need to find an equivalent angle.
To do this, we’ll just take away two 𝜋 from our value of 15𝜋 by four. And if we evaluate this expression, we get seven 𝜋 by four. What this means is, our angle of 15𝜋 by four is one full turn and then an extra turn of seven 𝜋 by four. In other words, what we’ve shown is, the point given to us in the question is actually equivalent to the polar coordinate point given by three, seven 𝜋 by four.
At this point, there are again a few different ways of finding where our angle of seven 𝜋 by four will be on the diagram. One way of doing this is to find seven 𝜋 by four divided by two 𝜋. This is our angle divided by two 𝜋, which is a full turn. This will tell us what proportion of a full turn our angle measures. If we evaluate this expression, we see that it’s equal to seven over eight. So our angle represents seven-eighths of a full turn. And we can find out where seven-eighths of a turn lies on our diagram very easily.
Each quadrant will be one-quarter of a turn. So we have one-quarter, two-quarters, three-quarters. And then, of course, seven-eighths is halfway between this quadrant. So our point must lie on this ray. We see that the only point is the point 𝐷. Therefore, our answer is the point 𝐷.
Therefore, we were able to show, of the five points given to us in the diagram, only the point 𝐷 is equivalent to the point with polar coordinates three, 15𝜋 by four.