The Unit Circle
- Craig Cullen
This worksheet shows a true unit circle display of the six trigonometric functions interpreted as geometric objects. The arc subtended by the angle is displayed in red around the circle and again translated into a dynamic line segment which represents the x-axis. On this page the sine function is labeled as the chord function to highlight the mistranslation of the word chord into the word sine. Additionally, the radius is displayed to help demonstrate that any circle can be considered a unit circle as long as the radius is taken as your unit. With this in mind the show/hide grid button displays a grid that is built on the size of the radius of the given circle. Along with this dynamic grid, the button show/hide radius units allows the user to display the radian units as they appear along the arc determined by the angle as well as the radian units along the "unwrapped" arc on the transformed arc. Each of the six trig functions can then be displayed as part of the unit circle as well as in the cartesian coordinate graph representation. Note that the secant and cosecant displays two different geometric interpretations each. In each case one display is vertical or horizontal, which makes determining when the function is positive/negative much easier. I recommend only displaying one of the six trigonometric functions at a time, and then pressing the play button in the bottom left corner to observe the connection of the two representations.
Using the geometric interpretation of the trigonometric functions produce geometric proofs for as many trigonometric identities as possible. You might start with the list below. tan(a) = chord(a)/cochord(a) Note in most textbooks this would be presented as: tan(a) = sin(a)/cos(a) chord(a) = a/cosecant(a) 1+secant^2(a) = tan^2(a)