Download Lecture Notes

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
-5
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.2
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 (identities):
sin x
1
tan x cos
cot x cos x 1
sec x cos
csc x 1
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 using the Quotient Rule.
2. Derivatives of sin x and cos x :
d Ÿsin x cos x
d Ÿcos x " sin x
and
dx
dx
Two facts are used in derivation: let 2 be in radians
and lim 1 " cos 2 0
lim sin 2 1
2
2
2v0
2v0
Derivative of sin x : Let fŸx sin x. Then
U
fŸx h " fŸx sinŸx h " sin x
lim
lim sin x cos h sinh cos x " sin x
f Ÿx lim
hv0
hv0
hv0
h
h
h
sin xŸcos h " 1 sin h cos x
sin h
lim
sin x lim cos h " 1 cos x lim
hv0
hv0
hv0
h
h
h
sin x Ÿ0 cos x Ÿ1 cos x
In the derivation of the derivative of cos x, you may need to use the identity:
cosŸx h cos x cos h " sin x sin h.
3. Derivatives of tan x, cot x, sec x, and csc x :
d Ÿcot x " csc 2 x,
d Ÿsec x tan x sec x,
d Ÿcsc x " cot x csc x
d Ÿtan x sec 2 x,
dx
dx
dx
dx
Derivations: Use the derivatives of sin x and cos x and the Quotient Rule
d Ÿtan x d sin x cos x cos x " sin xŸ" 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
0Ÿcos x " Ÿ1 Ÿ" sin x d Ÿsec x d
sin x
1
1
sin2x cos
x cos x tan x sec x
cos
x
dx
dx
cos x
cos 2 x
1
Example Let fŸx sin x and gŸx cos x. Find f Ÿ5 Ÿx , f Ÿ2003 Ÿx , g Ÿ5 Ÿx and g Ÿ2003 Ÿx .
n
f Ÿn Ÿx g Ÿn Ÿx 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
U
Example Find f Ÿx where
2
a. fŸx x
sin x
c. fŸx sec 2 x " tan 2 x
b. fŸx 2 sin x cos x
d. fŸx 2 sec 2 x
a.
2
U
f Ÿx 2x sin x "2x cos x 2x " x 2 cot x csc x
sin x
sin x
b.
U
f Ÿx 2¡cos 2 x sin xŸ" sin x ¢ 2 cos 2 x " sin 2 x
2 cosŸ2x c.
U
f Ÿx d Ÿ1 0
dx
d.
U
f Ÿx 2 d Ÿsec x sec x 2¡Ÿtan x sec x sec x sec xŸtan 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
U =
=
=
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
U
f Ÿx 2x cos x x 2 Ÿ" sin x 2x cos x " x 2 sin x
f
U
=
3
Ÿ2 =
3
cos =
3
"
=
3
2
sin =
3
2
the equation of the tangent line: y " = 18
2
= "
3
=
3
= "
3
=
3
2
3
2
2
3
2
x" =
3