Question Video: Converting Polar Coordinates to Cartesian Coordinates

Video Transcript

Two points in a plane have polar coordinates 𝑃 sub one 2.500 meters, 𝜋 over six and 𝑃 sub two 3.800 meters, two 𝜋 over three. Determine the Cartesian coordinates of 𝑃 sub one. Determine the Cartesian coordinates of 𝑃 sub two. Determine the distance between the points, to the nearest centimeter.

We can call the Cartesian coordinates of point 𝑃 sub one 𝑥 one, 𝑦 one and the Cartesian coordinates of 𝑃 sub two 𝑥 two, 𝑦 two. The distance between these two points we’ll call capital 𝐷. As starting information in this exercise, we’re given two points, 𝑃 sub one and 𝑃 sub two, in their polar coordinate setup. This means that the first coordinate in each pair is the radial distance. We’ll call that 𝑟 sub one for 𝑃 sub one and 𝑟 sub two for 𝑃 sub two. The second coordinate in the pair is the angular coordinate. In 𝑃 sub one, we’ll call that value 𝜃 sub one. And in 𝑃 sub two, we’ll call it 𝜃 sub two.

We can recall the coordinate conversion relationships between polar coordinates and Cartesian or 𝑥-, 𝑦-, 𝑧-coordinates. In our two-dimensional setup, if we take the polar coordinate 𝑟 and multiply it by the cosine of the angular polar coordinate 𝜃, then we’ll get the Cartesian coordinate 𝑥. Similarly, if we take that coordinate 𝑟 and multiply it by the sine of the angle 𝜃, we’ll get the Cartesian coordinate 𝑦.

This means that the Cartesian coordinates of the first point 𝑥 one, 𝑦 one are equal to 𝑟 sub one cos 𝜃 sub one and 𝑟 sub one sin 𝜃 sub one. When we plug these values in, using 2.500 meters for 𝑟 sub one and 𝜋 over six for 𝜃 sub one, we find a result of 2.165 meters in the 𝑥-direction and 1.250 meters in the 𝑦. These are the Cartesian coordinates of the point 𝑃 sub one.

Likewise, for 𝑥 two, 𝑦 two, where we’ll use polar values 𝑟 sub two and 𝜃 sub two. Using a value of 3.800 meters for 𝑟 sub two and two 𝜋 over three for 𝜃 sub two, we find a result of negative 1.900 meters in the 𝑥-direction and 3.291 meters in the 𝑦-direction. These are the Cartesian coordinates of the polar point 𝑃 sub two.

Finally, we want to solve for the distance 𝐷 between these two points. That distance, mathematically, is equal to the square root of the change in 𝑥 squared plus the change in 𝑦 squared. In our case then, 𝐷 is equal to the square root of 𝑥 two minus 𝑥 one quantity squared plus 𝑦 two minus 𝑦 one quantity squared. When we plug in our values for 𝑥 two, 𝑥 one, 𝑦 two, and 𝑦 one and then enter this expression on our calculator, we find that 𝐷 is 4.55 meters. That’s the distance between these two points, to the nearest centimeter.