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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.
y
y
1
0.5
1
0.5
0
0
0
1.25
2.5
3.75
5
6.25
0
1.25
2.5
3.75
5
6.25
x
x
-0.5
-0.5
-1
-1
y  sin x, 0 ≤ x ≤ 2
y  cos x, 0 ≤ x ≤ 2
y
1
0.5
0
-5
-2.5
0
2.5
5
x
-0.5
-1
— y  sin x, -.-. y  cos x, − 2 ≤ x ≤ 2
Relations (identities):
a.
sin x
1
i. tan x  cos
ii. cot x  cos x  1
iii. sec x  cos
iv. 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.
b. More commonly used identities:
ii. sec 2 x − tan 2 x  1
i. sin 2 x  cos 2 x  1
iii. sin2x  2 sin x cos x
iv. cos2x  cos 2 x − sin 2 x  2 cos 2 x − 1  1 − 2 sin 2 x
c. Sum-difference formulas:
i. sina  b  sina cosb  sinb cosa
ii. cosa  b  cosa cosb ∓ sina sinb
1
Values of Trigonometric Functions at Special Angles:
x in radians (degrees)
sin x
cos x
tan x
cot x
csc x
sec x
00 ∘ 
0
1
0


1
 30 ∘ 
6
1
2
3
2
1
3
3
2
2
3
 45 ∘ 
4
1
2
1
1
2
2
 60 ∘ 
3
 90 ∘ 
2
3
2
1
2
3
1
3
2
3
2
1
0

0
1

1
2
2. Derivatives of sin x and cos x :
d cos x  − sin x
d sin x  cos x
and
dx
dx
Two facts are used in derivation: let  be an angle in radians
lim sin   1 and lim 1 − cos   0
→0
→0


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 will 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 csc x  − cot x csc x
d cot x  − csc 2 x,
d sec x  tan x sec 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
2
2
2
dx
dx cos x
cos x
cos x
cos x
0cos x − 1− sin x
sin x
d sec x  d
1
1

 sin2x  cos
x cos x  tan x sec x
cos
x
dx
dx
cos x
cos 2 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 Compute f x where
2
b. fx  2 sin x cos x
a. fx  x
sin x
c. fx  sec 2 x − tan 2 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 sec 2 x − tan 2 x  d 1  0
dx
dx
d.
′
f x  2 d sec x sec x  2 tan x sec x sec x  sec xtan x sec x
dx
 4 tan x sec 2 x
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
f
x−
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
2

3
3
2
2
3
2
x− 
3