Derivative of Trigonometric Functions

 

Motivation:

                         We start to find the derivative of the function sin(x) from definition:

 

                         sin(x+h)-sin(x) = sin(x) cos(h) + cos(x) sin(h) – sin(x)

                                                  = sin(x)( cos(h) -1 ) + cos(x) sin(h)

 

                        (sin(x+h)-sin(x))  = sin(x)( cos(h)-1 )  + cos(x) sin(h)

 

                         Then,  we try to find the limit by considering  h  0 .  However, we encounter hurdle here:  what is the value of the term

 

                                                      ???

 

                          We do not even know if the limit exists .   So, it is necessary to explore this problem at first.

 

 

Theorem:   

 

          Proof:

                             Recall the definition of  sin .   Let O be the center of unit circle as shown below.  ,   BOA=  ,   ,  and  .

 

                                              

 

                           Thus,  while  < ,  we have

 

                                         arc AB =        = sin

 

                          and   tan  =   arc AB 

                             ( This can be  obtained via the comparison of area )

                      

                          So,

                                     tan     sin

                                     sin

                                   1

                           While ,    1.

 

                          Hence, 

 

 

Derivative of sin(x)

 

                              Let’s continue finding the derivative of sin(x) :

 

                        (sin(x+h)-sin(x))  = sin(x)( cos(h)-1 )  + cos(x) sin(h)

 

                         sin’(x) =

                                     =  +

                                     = 0 + cos(x)

                                     = cos(x)

 

                      With this result, we can quickly summarize the following things.

 

 

Theorem:  (1)  sin’(x) = cos(x)

                   (2) cos’(x) = -sin(x)

                   (3) tan’(x) = sec²(x)

                   (4) sec’(x) = sec(x)tan(x)

 

 

           Proof:

(1)     is done via the previous result.  Let’s start from (2).

(2)       

          cos(x) = sin(  -x)

           By chain rule, we have

 

                   cos’(x) = cos(  -x)(-1) = -sin(x)

(3)     tan(x) =

So,

         tan(x) =

                   =

                   =

 

(4)     sec(x) =

sec’(x) =  = sec(x)tan(x)