Math 221 
                               Week 4 part 2 
                  Trigonometric functions: 
                      limits and derivatives
                                                                                                             Please take a moment to just breathe.
                                                                                                    We define sinθ, cosθ, and tanθ using the unit circle.

                                                                                                    Note that by similar triangles, tanθ = sinθ 

               In this section:
                                                                                                      1      cosθ
               We review the trigonometric functions.

               We prove that lim sinh = 1.

                               h→0    h
               We derive the derivative of several trigonometric functions.

                                                                                                                                1    sinθ  tanθ
                                                                                                                           θ     cosθ
                                            We will show that 
                                                                                                                                                                                                                                                To compute the derivatives, we will need the following limits.

                                                                       d                                                                                                                                                                                                                                        lim sinh = 1                            lim cos h − 1 = 0

                                                         dt sin t = cos t .
                                                                                                                                                                                                                                   h→0                  h                                                                                 h→0                        h
                                                                                                                                                                                                                                                                                                               As evidence of the first limit, recall that the arc of a unit circle 
                                                                                                                                                                                                                                                                                                               of angle θ has length θ. The line segment representing sinθ

                                                                                                                                                                                                                                                                                                                                                                                         appears to have nearly the 

                                            position of the mass:    
                                                                                                                                                                                                                                                                                                                                                     same length as the bit of arc. 

                                              p(t) = sin t
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              

                                            velocity of the mass:     

                                             p′( t) = cos t

                                             Proof using Squeeze Theorem that  lim sinh = 1.      
                                                                                                                                                                                   +          h
                                                                                                                                                                       h→0
                                              For h in the interval [0, π/2],   
                                                                                                                                                                                                                            The proofs that

                                             

                                             sinh ≤ h ≤ tanh,   so 
                                                                                                                                                                                                                                              lim sinh = 1         and        lim cosh − 1 = 0

                                                                  h                             1                                                                                                                                                                                                                 h→0−                   h                                                                        h→0                         h
                                             1 ≤ sinh ≤ cosh
                                                                                                                                                                                                                                                 are similar.

                                             By the Squeeze Theorem, since  lim                                                                                              1             =1, 
                                                                                                              We will also need these formulas: 

                                                                     h                                                                                h→0+ cosh
                                               lim                                =1.
                                                                                                                                                                                                                        sin(x + h) = sin xcosh + cosxsinh

                                                          + sinh
                                              h→0                                                                                                                                                                                                                                                              cos(x + h) = sin xsinh − cosxcosh

                                                                                                                                                                                                                                                                                                               We have

                                            Use the limit definition to compute the derivative of sin                                                                                                                         x:
                                                                                  d sinx = cosx            

                                                d                                                 sin(x + h) − sin x                                                                                                                                                                                              dx
                                             dx sinx = lim                                                                   h                                
                                                                                                                                               and a similar computation shows   

                                                                                    h→0
                                              =lim sinxcosh+cosxsinh−sinx
                                                                                                                                                                                                                                      d cosx = −sinx

                                                       h→0                                                              h                                                                                                                                                                                       dx
                                              =lim sinxcosh−sinx+cosxsinh
                                                                                                                                                                                                                                    Exercise:  compute  d tan x using the quotient rule.

                                                       h→0                                                              h                                                                                                                                                                                                                                                       dx
                                              =lim[sinxcosh−1 +cosxsinh] = cosx                                                                                                                                                                                                                               (Please pause the video and try it yourself!)

                                                       h→0                                            h                                                     h
                                                     
                                                                                                                                                                                                                                                         Trigonometric functions and their derivatives.  
                                               d tanx = d [sinx]

                                              dx                                     dx cosx                                                                                                                                                                                                                   sin x          cos x

                                                                                                                                                                                                                                                                                                               tan x          sec2 x

                                              = (sinx)′( cosx) − (sin x)(cosx)′
                                                                                                                                                                                                                            secx          sec x tan x

                                                                                                 cos2x                                                                                                                                                                                                         cosx         −sin x

                                                                                                                                                                                                                                                                                                               cot x         −csc2 x

                                              = (cosx)(cosx)−(sinx)(−sinx)
                                                                                                                                                                                                                                   cscx         −csc x cot x

                                                                                                  cos2x
                                                                                                                                                                                                                                                                                                               If a trigonometric function starts with “co”, then its derivative 
                                                                               2                                 2                                                                                                                                                                                             has a negative sign.

                                              = (cosx) +(sinx) =                                                                        1               =sec2x                                                                                                                                                Please memorize these.
                                                                            cos2x                                                cos2x
            A side note about  lim sinh = 1                           
                            h→0  h
            In practice, the approximation sinh ≈ h for small h is very 
            useful.

            If we want to speed up the oscillation, we also have

            lim sin(2h) = 1,   etc.

            h→0   2h
            Here is the full “Taylor series” for sin x from Calc II.

            sinh = h−h3/3!+h5/5!−h7/7!−h9/9!+....

            The more terms you take, the better the approximation will be.