Extracting x and y values from radians

Question

Let's assume I have the formula $x^2+y^2=r^2$; This will produce a circle when put into a graphing calculator. Now, let's assume I am given a value $a$, for radians. How can I extract the value that $x$ and $y$ can be using radians? I know it is possible - I am just not sure how to formulate this.

Answer 1

Radians are units used for measuring angles. One presumes that you are trying to find the point on the unit circle that corresponds to an angle of $a$ radians.

That is, draw ray originating at the origin that makes an angle of $a$ radians to the positive $x$-axis, measuring anti-clockwise. This ray will intersect the circle of radius $r$ at some point. The coordinates of that point will be $$ (x_a, y_a) = (r \cos(a), r\sin(a)).$$

This follows from the definition of the sine and cosine functions. To wit, if we draw the ray described above, it will intersect the unit circle, i.e. the circle of radius 1 centered at the origin which is given by the equation $$ x^2 + y^2 = 1,$$ at some point. By definition, we say that the cosine of $a$ is the $x$-coordinate of that point, and that the sine of $a$ is the $y$-coordinate of that point.

To get the point on the circle with radius $r$, dilate by a factor of $r$. This will multiply every length by $r$. In particular, the $x$-coordinate of the intersection goes from $\cos(a)$ to $r \cos(a)$, and the $y$-coordinate of the intersection goes from $\sin(a)$ to $r \sin(a)$.

Answer 2

There's a pretty easy way to see this. Draw a right triangle with one vertex at the origin, one vertex on the circle, and one vertex on the $x$-axis, where the angle at the origin is $a$ and the right angle is the other angle on the $x$-axis. The hypotenuse is a radius of the circle, so its length is $r$. Basic trigonometry tells you that the length of the side on the $x$-axis is $r\cos a$ and the length of the other leg is $r\sin a$. These are your coordinates!