This worksheet shows a true unit circle display of the six[br]trigonometric functions interpreted as geometric objects. The arc subtended by[br]the angle is displayed in red around the circle and again translated into a[br]dynamic line segment which represents the x-axis. [br][br][br]On this page the sine function is labeled as the chord[br]function to highlight the mistranslation of the word chord into the word sine.[br]Additionally, the radius is displayed to help demonstrate that any circle can[br]be considered a unit circle as long as the radius is taken as your unit. With[br]this in mind the [b]show/hide grid[/b] button displays a grid that is built on[br]the size of the radius of the given circle. Along with this dynamic grid, the[br]button [b]show/hide radius units[/b] allows the user to display the radian[br]units as they appear along the arc determined by the angle as well as the[br]radian units along the "unwrapped" arc on the transformed arc. [br][br]Each of the six trig functions can then be displayed as part[br]of the unit circle as well as in the cartesian coordinate graph representation.[br]Note that the secant and cosecant displays two different geometric[br]interpretations each. In each case one display is vertical or horizontal, which[br]makes determining when the function is positive/negative much easier. I[br]recommend only displaying one of the six trigonometric functions at a time, and[br]then pressing the play button in the bottom left corner to observe the[br]connection of the two representations.[br][br][br]

Using the geometric interpretation of the trigonometric[br]functions produce geometric proofs for as many trigonometric identities as[br]possible. You might start with the list below. [br][br]tan(a) = chord(a)/cochord(a)   Note in most textbooks this would be[br]presented as: tan(a) = sin(a)/cos(a)[br][br]chord(a) = a/cosecant(a)[br][br]1+secant^2(a) = tan^2(a)[br][br][br][br][br][br][br]