Download 2.5 Derivatives of Trigonometric Functions 1. Six Trigonometric

2.5 Derivatives of Trigonometric Functions
1. Six Trigonometric Functions and Identities:
Six trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x, x is in radians
Recall: Functions sin x and cos x are periodic functions with period 2!. They are continuous everywhere
and graphically they are also differentiable everywhere. Their derivatives exist everywhere.
-10
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.2
-5
5 x
10
-10
-5
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
5 x
10
y ! cos x
y ! sin x
Relations:
1
sin x
cot x ! cos x ! 1
sec x ! cos
csc x ! 1
tan x ! cos
x
x
tan x
sin x
sin x
So, if we know the derivatives of sin x and cos x, then we can derive the derivatives of tan x, cot x, sec x and
csc x by the Quotient Rule.
2. Derivatives of sin x and cos x :
d !sin x" ! cos x
dx
Recall: Two facts - " in radians
lim sin " ! 1
"
""0
d !cos x" ! ! sin x
dx
and
and
lim 1 ! cos " ! 0
"
""0
1.5
0.4
1
0.2
0.5
-1.5
-1
-0.5
0
0.5
x1
-0.5
1.5
-0.4
-0.2
0
0.2 x
-0.2
-1
-1.5
– y ! sin x, -.-. y ! x
1
-0.4
– y ! sin x, -.-. y ! x
0.4
1.5
0.4
1
0.2
0.5
-1.5
-1
-0.5
0
-0.5
0.5
x1
1.5
-0.4
-0.2
0
0.2 x
0.4
-0.2
-1
-1.5
-0.4
– y ! tan x, -.-. y ! x
– y ! tan x, -.-. y ! x
Proof (1) Graphically, sin x " x, and tan x # x. So, sin x " x " tan x.
sin x
sin x
x
sin x " x iff sin
x " 1; x " tan x iff x " cos x iff cos x " x
Hence,
x
cos x " sin
x " 1.
Because lim x"0 cos x ! 1, and lim x"0 1 ! 1, by the Squeeze Theorem we
x
lim sin
x ! 1.
x"0
(2) Recall: sin 2 " # cos 2 " ! 1, or 1 ! cos 2 " ! sin 2 "
!1 ! cos 2 ""
!1 ! cos "" !1 # cos ""
lim 1 ! cos " ! lim
! lim
"
"
""0
""0
""0 "!1 # cos ""
!1 # cos ""
2
sin "
1
! lim
! lim sin " sin "
! !1"!0" 1 ! 0
"
""0 "!1 # cos ""
""0
2
!1 # cos ""
Derivative of sin x : Let f!x" ! sin x. Then
$
f!x # h" ! f!x"
sin!x # h" ! sin x
! lim
! lim sin x cos h # sinh cos x ! sin x
f !x" ! lim
h"0
h"0
h"0
h
h
h
sin x!cos h ! 1" # sin h cos x
sin h
! lim
! sin x lim cos h ! 1 # cos x lim
h"0
h"0
h"0
h
h
h
! sin x !0" # cos x !1" ! cos x
In the derivation of the derivative of cos x, you may need to use the identity:
cos!x # h" ! cos x cos h ! sin x sin h.
3. Derivatives of tan x, cot x, sec x, and csc x :
d !cot x" ! ! csc 2 x,
d !sec x" ! tan x sec x,
d !csc x" ! ! cot x csc x
d !tan x" ! sec 2 x,
dx
dx
dx
dx
Derivations: Use the derivatives of sin x and cos x and the Quotient Rule
d !tan x" ! d sin x ! cos x cos x ! sin x!! sin x" ! cos 2 x # sin 2 x ! 1
! sec 2 x
dx cos x
dx
cos 2 x
cos 2 x
cos 2 x
0!cos x" ! !1"!! sin x"
d !sec x" ! d
sin x
1
1
!
! sin2x ! cos
x cos x ! tan x sec x
dx
dx cos x
cos 2 x
cos x
Example Let f!x" ! sin x and g!x" ! cos x. Find f !5" !x", f !2003" !x", g !5" !x" and g !2003" !x".
2
n
f !n" !x"
g !n" !x"
0
sin x
cos x
1
cos x
! sin x
2
! sin x
! cos x
3
! cos x
sin x
4
sin x
cos x
5
cos x
! sin x
2003 ! cos x
sin x
$
Example Find f !x" where
2
a. f!x" ! x
sin x
c. f!x" ! sec 2 x ! tan 2 x
b. f!x" ! 2 sin x cos x
d. f!x" ! 2 sec 2 x
a.
2
$
f !x" ! 2x sin x !2x cos x ! 2x ! x 2 cot x csc x
sin x
sin x
b.
$
f !x" ! 2#cos 2 x # sin x!! sin x"$ ! 2 cos 2 x ! sin 2 x
! 2 cos!2x"
c.
$
f !x" ! d !1" ! 0
dx
d.
$
f !x" ! 2 d !sec x sec x" ! 2#!tan x sec x" sec x # sec x!tan x sec x"$ ! 4 tan x sec 2 x
dx
Example Find the equation of the tangent line to the curve y ! x 2 cos x at a ! ! .
3
$ !
!
!
the equation of the tangent line: y ! f
x!
!f
3
3
3
2
2
1 ! !2
f ! ! !
cos ! ! !
18
3
3
3
3
2
$
f !x" ! 2x cos x # x 2 !! sin x" ! 2x cos x ! x 2 sin x
f
$
!
3
! !2" !
3
cos !
3
!
!
3
2
sin !
3
2
the equation of the tangent line: y ! ! !
18
3
! !
3
!
3
! ! !
3
!
3
2
3
2
2
3
2
x! !
3