The Pythagorean Theorem - Educational Outreach

The six trigonometric functions are defined as follows:1] ) sin( t) ysin( 2]csc( ) csc( t)1y3] ) cos( t) xcos( 4]1sec( ) sec( t)x5]ytan( ) tan( t) 6]xcot() cot( t)xyAs the independent variable t increases, one eventuallyreturns to the point ( 1,0)and circles around the rim asecond time, a third time, and so on. Additionally, any fixedpoint x , ) on the rim is passed over many times as we( y 0 0continuously spin around the circle in a positive or negativedirection. Let t0or 0be such thatr , one completeP( t0 0 0y0) P( ) ( x , ) . Since 1revolution around the rim of the circle is equivalent tounits and, consequently,P t ) P(t 2n ) : n 1,2,3... . We also have that(0 00( 0) P(3600) sin( 00t0 ) sin( t 2n) sin( 0) sin(360 0P . For )2t , this translates tosin( 0or ). Identicalbehavior is exhibited by the five remaining trigonometricfunctions.In general, trigonometric functions cycle through the samevalues over and over again as the independent variable tindefinitely increases on the interval [ 0, ) or, by the sametoken, indefinitely decreases on the interval (,0],corresponding to repeatedly revolving around the rim of theunit circle in Figure 3.15.119