Formula of Sine and Cosine for Sum of the angles

 

 

Motivation:       Let    AOB =   in the following diagram.

                        A and B are on the unit circle. Thus, their coordinates can be

                        denoted as (sinA, cosA) and  (sinB, cosB) correspondingly.

                                                                       

                                           

 

                             Recall the result mentioned for the inner product of  vectors,

                                                 cos  =     ( it is via cosine law )

 

                             Thus,            cos  = sinA sinB + cosA cosB

                             Please notice that   is the difference between the two angles spanned by A and B.

 

                                                 cos(A-B) =  cosA cosB + sinA sinB

 

                             Hence, we can derive the formulae for trigonometric functions associated with sum of angles.

 

 

 

 

 

 

 

 

Properties:         1.   cos(A-B) = cosA cosB + sinA sinB

2.      cos(A+B) = cosA cosB – sinA sinB

3.      sin(A+B) =  sinA cosB + cosA sinB

4.      sin(A-B) = sinA cosB – cosA sinB

 

             Proof:

                         

                            From the introduction above, we have

                                         cos(A-B) = cosA cosB + sinA sinB .

 

                           The rest of them are straightforward:

 

                                    cos(A+B) =  cos(A-(-B))

                                                    =  cosA cos(-B) + sinA sin(-B)

                                                    = cosA cosB – sinA sinB

 

                                    sin(A+B) = cos( 90^o – (A+B))

                                                   = cos( (90^o – A) – B )

                                                   = cos(90^o-A) cosB + sin(90^o – A)sinB

                                                   = sinA cosB + cosA sinB

 

                                    sin(A-B) = sin(A+(-B))

                                                  = sinA cos(-B) + cosA sin(-B)

                                                  = sinA cosB – cosA sinB

 

 

 

 

 

 

 

 

Property:                         tan(A+B) = 

 

                Proof:

 

                                  tan(A+B) = 

                                                 =  

 

                                    the numerator and denominator can be divided

                                    by cosA cosB at the same time :

 

                                tan(A+B) = 

 

                                                                       

 

 

 

 

 

 

 

 

 

 

                   If we set A=B, we can get the following equalities.

 

 

Property:   sin(2A) = 2sinA cosA

                   cos(2A) = cos²A – sin²A

                   tan(2A) =