Mat h 12 05 Ca lc ul us /S ec. 3 .4 De riv ative s o f T ri gono met ric F uncti o ns
I.
Special Trigonometric Limits
" sin(t ) %
lim $
' =1
t!0 #
t &
II.
# cos(t ) " 1&
lim %
(=0
t!0 $
'
t
Derivatives of Trigonometric Functions
A. Derivatives of Trigonometric Functions
1.
d
(sin x ) = cos x
dx
4.
d
(csc x ) = ! csc x cot x
dx
2.
d
(cos x ) = !sin x
dx
5.
d
(sec x ) = sec x tan x
dx
3.
d
(tan x ) = sec 2 x
dx
6.
B. Proofs
1. Proof of the derivative of
d
(cot x ) = ! csc 2 x
dx
sin(x) .
d
# sin ( x + h ) " sin(x) &
(sin(x)) = lim
('
h!0 %
$
dx
h
Using the trig identity sin ( a + b ) = sin a cosb + sin b cos a yields
d
# sin(x)cos(h) + sin(h)cos(x) " sin(x) &
(sin(x)) = lim
%
('
h!0 $
dx
h
# sin(x)cos(h) " sin(x) + sin(h)cos(x) &
= lim %
('
h!0 $
h
# sin(x) ( cos(h) " 1) + sin(h)cos(x) &
= lim %
('
h!0 $
h
#
( cos(h) " 1) + sin(h) cos(x)&
= lim % sin(x)
('
h!0 $
h
h
#
( cos(h) " 1) & + # lim sin(h) &
= lim sin(x) % lim
%$ h!0
(
('
h!0
$ h!0
h
h '
(
)
= sin(x) ( 0 ) + (1) cos(x)
d
(sin(x)) = cos(x)
dx
**See the proof of
cos(x) in your test, p158-159.
(lim cos(x))
h!0
2. Prove that
d
(tan x ) = sec 2 x
dx
C. Examples: Differentiate the following.
6
x
1.
y = cot(x) + 3x ! sec(x) +
2.
f (t) = t 2 sin(t)
3.
! ex $
&& Differentiate 2 different ways
g( x ) = ##
" tan x %
3
4.
5.
" sin ! + cos! %
r=$
'
#
cos ! &
y=
cos x
1 ! sin x
6. Find the equation of the tangent line to the curve
y = 3 + 12 x + sin x at the pt (0,3).
7. Determine the point(s) where
y = cos x + 3 sin x has horizontal tangent(s).
"x + k ; x ! 0
g(x) = #
$cos x ; x > 0
a. Is there a value of k that will make g(x) continuous at x=0?
7. Given
b. Is there a value of
k that will make g(x) differentiable at x=0?