Double-Angle and Half-Angle Identities

                         Double-Angle Identities

Letting in the identity
     
we get the

          double-angle identity for the sine function
                    

Similarly, letting in the identity
     
we get the
                    first double-angle identity for the cosine function
                         

Using the Pythagorean identity
     
we get the
                    second double-angle identity for the cosine function
                         
and the
                    third double-angle identity for the cosine function,
                         

Since the tangent is the ratio of the sine to the cosine, and using the above
identities, and some simple algebra, we get the various forms of the

                    double-angle identity for the tangent function

                         


See Examples 1 – 2, pages 544 – 545, of the textbook.



                              Half-Angle Identities

The half-angle identities are just the double-angle identities given in a different form.

  can be written as   

For example, the double-angle identity

     

can be rewritten as

     
                              or

     

Solving this equation for  , we get the

          Half-Angle Identity for the Sine Function

                    
where the sign,    or   , is determined by the quadrant in which    lies.

Similarly we get the

          Half-Angle Identity for the Cosine Function

               
where the sign,    or   , is determined by the quadrant in which      lies.

Since the tangent is the ratio of the sine to the cosine, we get the

          Half-Angle Identity for the Tangent Function

               
where the sign,    or    , is determined by the quadrant in which   lies.

Other half-angle identities for the tangent function are

          
and

          


     See Examples 3 – 5, pages 548 – 549, of the textbook


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