The Unit Circle
The Unit Circle
The Unit Circle is a circle with a radius of 1 centered at the point (0,0). Moving the green angle slider will rotate the point at (1,0) around the circle in a counterclockwise direction. Together with the point at the center of the circle that rotating point forms a right triangle. As we move the point between 0 and 90 degrees, the sine and cosine of the angle of rotation are simply the y and x values of the rotating point. This is true because the hypotenuse for these triangles is just the radius of the circle, which is always 1. Therefore, for example, the ratio for the sine, opposite/hypotenuse is equal to the opposite/1 or just the length of the side opposite the angle. This is equivalent to the y-coordinate of the rotating point. As move into the other quadrants, the angle of rotation is no longer inside the triangle but the relationship between the coordinates of the rotated point and the sine and cosine of the angle of rotation remains true: for a rotation of 135, for example, the x- and y-values of the rotated point are each , which is also the sine and cosine of 135. When the rotated point reaches any of the marked points, which are all the multiples of 30 and of 45, the sine and cosine values are exact because the triangle formed is a special right triangle, either 30-60-90 or 45-45-90. By checking the coordinate and the angle measure checkboxes, you can see the coordinates of all of those points as well as the angle of rotation they represent, expressed in degrees and in radians.