Here are some formulae that you may find useful in differentiating the trigonometric functions and in
other areas of your life:
sin(a  b)  sin(a) cos(b)  sin(b) cos(a)
cos(a  b)  cos(a) cos(b) sin(a) sin(b)
 sin x 
lim 
1
x 0  x 

 cos x  1 
lim 
0
x 0 
x 
sin  2   2sin( )cos( )
17  17  34
(1)
Derive the formulae for the derivatives of sin( x) and cos( x) using the limit definition of the
derivative.
Dx  sin x  
Dx  cos x  
(2)
Differentiate each of the following using the rules that we have developed for differentiation.
.
f  x   sec( x)
h  x   tan( x)
g  x   cot  x 
f  x   csc( x)
yper
(3) Differentiate the following. You need not simplify the result.
j  x 
cot  x 
x2
k ( x)   sin x  cos  x  
2
(4) Differentiate the hyperbolic trig functions.
f ( x)  cosh x
g ( x)  sinh x
(5) Define the four other hyperbolic trig functions and find their derivatives.
(6) For fun – we don’t usually work in the complex numbers in calculus. Remember that
cis()  ei  cos   i sin  . Use this identity to find a formula for cos(i) .