Course II 
                                                 
                                      Lesson ４	
   Derivatives of Trigonometric Functions 　	
                                                 
                                      ４A 
                    •  Derivative of Sine Function 
                    •  Limit of    sinx
                                     x
                    •  Derivatives of Basic Trigonometric Function 
                                                                                         1 
                 Derivative of Sine Function 
   Variation of slopes          Cosine ? 
                 f (x)
                                              f (x) = sin x
   Derivative by definition 
                    ʹ         f (x +h)− f (x)    sin(x+h)−sinx
                   f (x) = lim              =lim
                          h→0      h          h→0       h
                 =limsinxcosh+cosxsinh−sinx
                   h→0            h
                       ⎛cosxsinh −sin x1−cosh⎞
                 =lim⎜                       ⎟
                  h→0  ⎝      h          h   ⎠                 2 
                                      sin x/ x
                                       Limit of  
    Consider a sector with central angle   
                                      x
    Compare the areas of △OAB, sector OAB, and △OAT  
             1               2   x   1
             2⋅1⋅sinx<(π ⋅1 )⋅ 2π < 2⋅1⋅tanx
            ∴ sinx 0)
                 ∴ 1< x < 1
                        sinx  cosx                      y = x
                        sinx                                  y =sin x
                 ∴ 1> x >cosx
      As  x→0                   1
                  limsinx =1
                   h→0  x                                           3 
               Derivative of Sine FunctionーCont. 
       ʹ        ⎛      sin h      1−cosh⎞
      f (x) = lim cosx      −sin x
             h→0⎜       h            h    ⎟
                ⎝                         ⎠
                        １	 (1−cosh)(1+cosh)     1−cos2h    sinh  sinh
                                 h(1+cosh)    = h(1+cosh) = h (1+cosh)
                                                            １	    ０	
        Therefore 
                         ʹ
                   (    )
                    sinx =cosx
                                              That makes sense!            4