Trigonometrics – Definition of Trigonometric Functions on a Unit Circle

 

 

Motivation:   Consider the Unit Circle ( radius=1 ) on X-Y plane in the

                       following diagram. Let A be a sliding point moving on this unit circle.

                       The measure of the angle between  and  ( positive direction

                       along x-axis )  can be considered as the arc length from D to A.

                                                                       

                                                       

                    If  A is moving counterclockwise, we say the angle is increased; 

                    if A is moving clockwise, we say the angle is decreased. 

                    We denote the angle as   .

 

                The notion of  the trigonometric functions we define for acute angles

                can be generalized; and the argument of those functions does not have

                to be limited to acute angles any more if we think in the following way:

 

                            sin  :  the y-coordinate of point A

                            cos  : the x-coordinate of point A

                            tan  = sin  /cos    ( that is equal to   )

                            csc  = 1/sin

                            sec  = 1/cos

                            cot  = 1/tan

 

 

                It does not contradict to the definition for the cases in acute angles.   The official definition is as follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

Definition:       For the following diagram,  the angle between positive direction of x-axis and  is denoted as .

 

                                           

                      The associated trigonometric functions  are defined as follows:

 

 

                                   sin  = 

                                   cos  = 

 

                                   tan  =

                                   csc  =                                    

                                   sec  =

                                   cot  =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Properties:         

                  1.      sin(180^o -  ) = sin

                           cos(180^o­ -   ) = -cos

                  2.      sin(90^o -    ) = cos

                           cos(90^o -    ) = sin   

3.            sin(-  ) = -sin

4.            cos (-  ) = cos

 

 

    

 

 

                         The relationship can be shown by using some diagrams like below:

                                                     

                                                      

 

 

 

 

 

 

 

 

 

 

 

 

Theorem ( Cosine Law )

 

                             

                 Recall the cosine law we have for the diagram on the left.  It can be shown that the relationship also holds for the diagram on the right by using   cos(180^o­ -   ) = -cos  .

 

                 Thus, we have

 

                                         cosB =   (  ² +  ² -  ² )

 

                         If we use the convention that the side corresponding to the associated angle is denoted by  its corresponding lowercase letter,  it can be rewritten as

 

                                         cosB =   ( a² + c² – b² )

 

 

                                                                       

 

Theorem ( Sine Law )

                                  

                                  

 

                 Proof:

                                

                                                   

                                      

                                   

                        For the diagram shown on the left, sinB = sin(180⁰ –B).  Thus,

                         =  bc sinA =   ba sinC =  ca sinB

 

                        Let each be divided  by abc:

 

                                            =  =

                        Thus,