Given that the polar coordinates of point 𝐴 are four, 120 degrees, find the Cartesian coordinates of 𝐴.
First, let’s sketch a graph where point 𝐴 would be located. Polar coordinates mark how far away and at what angle we would find point 𝐴. We would find point 𝐴 at 120 degrees from the origin like this. It is also four units from the origin, point 𝐴. To mark the Cartesian coordinates of point 𝐴, we need to know its location along the 𝑥- and 𝑦-axis.
And we do that by creating a right triangle to help us calculate side lengths 𝑎 and 𝑏 like this. Can we find out what angle would be created here? Since we know that half a turn is 180 degrees, we can say that the angle created inside this triangle is a 60-degree angle. We know an angle measure and the hypotenuse of this triangle. And that means we can use sine or cosine to solve for the missing side length.
To find side length 𝑎, it’s the side opposite the angle we have. sine of 60 degrees equals the measure of side length 𝑎 over the hypotenuse of four. Side length 𝑏 is the adjacent side length to our angle. It can be found by calculating the cosine of 60 degrees which is equal to the missing side length 𝑏 over four.
To solve for 𝑎, we multiply the right side by four and the left side by four. 𝑎 equals four times sine of 60 degrees. We do the same thing for side length 𝑏. Multiply both sides by the hypotenuse four and 𝑏 equals four times the cosine of 60 degrees. Four times the sine of 60 degrees equals two times the square root of three. Our 𝑎 value, our 𝑎 side length, is two times the square root of three units long. Four times cosine of 60 degrees equals two. And that means the measure of our 𝑏 side is two units.
But how do we take these values and turn them in to coordinates? Side length 𝑏 is two units. And that means that one of the vertexes of our triangle falls at negative two on the 𝑥-axis. Since our triangle has a height of two times the square root of three, it is located at the point two times the square root of three on the 𝑦-axis.
Another way to check that our signs are correct is to know that point 𝐴 falls in quadrant two. And quadrant two is the location of negative 𝑥-values and positive 𝑦-values.
Point 𝐴 is located at negative two on the 𝑥-axis and positive two times the square root of three on the 𝑦-axis.