The Pythagorean Theorem - Educational Outreach

The following four relationships are immediatelyevident from the above six definitions. These fourrelationships will comprise our primitive trigonometricidentities.1]3]1csc( ) 2]sin( )sin( )tan( ) 4]cos( )1sec( )cos( )cos( )cot( )sin( )Since the four trigonometric functions csc( ) , sec( ) ,tan( ) , and cot( ) reduce to nothing more thancombinations of sines and cosines, one only needs to knowthe value of sin( ) and )cos( for a given in order toevaluate these remaining four functions. In practice,trigonometry is accomplished by table look up. Tables forthe six trigonometric function values versus are set up00for angles from 0 to 90 in increments of 10 ' or less.Various trigonometric relationships due to embeddedsymmetry in the unit circle were used to producetrigonometric functional values corresponding to angles00ranging from 90 to 360 . Prior to 1970, these values weremanually created using a book of mathematical tables.Nowadays, the painstaking operation of table lookup istransparent to the user via the instantaneous modernelectronic calculator.Our second pillar is a group of three identities,collectively called the Pythagorean identities. ThePythagorean identities are a direct consequence of thePythagorean Theorem. From the Pythagorean Theorem andthe definitions of sin(t)and cos(t ) , we immediately seethat22sin ( t ) cos ( t) 1 .121