3.5: Derivatives of Trigonometric...
Transcript of 3.5: Derivatives of Trigonometric...
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Part 1
3.5: Derivatives of Trigonometric Functions
Part 1: The Sine and Cosine
MATH 165: Calculus I
Department of Mathematics
Iowa State University
Paul J. Barloon
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Needed Trigonometric Identities
To find the derivatives of sin(x) and cos(x), we first need toreview two trigonometric identities:
sin(x + h) = sin(x) cos(h) + cos(x) sin(h)
cos(x + h) = cos(x) cos(h)� sin(x) sin(h)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Needed Trigonometric Identities
To find the derivatives of sin(x) and cos(x), we first need toreview two trigonometric identities:
sin(x + h) = sin(x) cos(h) + cos(x) sin(h)
cos(x + h) = cos(x) cos(h)� sin(x) sin(h)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Needed Trigonometric Identities
To find the derivatives of sin(x) and cos(x), we first need toreview two trigonometric identities:
sin(x + h) = sin(x) cos(h) + cos(x) sin(h)
cos(x + h) = cos(x) cos(h)� sin(x) sin(h)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of sin(x)
To find the derivative of f (x) = sin(x), we now use the definition:
f
0(x) = limh!0
sin(x + h)� sin(x)
h
= limh!0
[sin(x) cos(h) + cos(x) sin(h)]� sin(x)
h
= limh!0
sin(x)[cos(h)� 1] + cos(x) sin(h)
h
= limh!0
sin(x)[cos(h)� 1]
h
+ limh!0
cos(x) sin(h)
h
= sin(x) · limh!0
cos(h)� 1
h
+ cos(x) · limh!0
sin(h)
h
= sin(x)(0) + cos(x)(1) = cos(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of sin(x)
To find the derivative of f (x) = sin(x), we now use the definition:
f
0(x) = limh!0
sin(x + h)� sin(x)
h
= limh!0
[sin(x) cos(h) + cos(x) sin(h)]� sin(x)
h
= limh!0
sin(x)[cos(h)� 1] + cos(x) sin(h)
h
= limh!0
sin(x)[cos(h)� 1]
h
+ limh!0
cos(x) sin(h)
h
= sin(x) · limh!0
cos(h)� 1
h
+ cos(x) · limh!0
sin(h)
h
= sin(x)(0) + cos(x)(1) = cos(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of sin(x)
To find the derivative of f (x) = sin(x), we now use the definition:
f
0(x) = limh!0
sin(x + h)� sin(x)
h
= limh!0
[sin(x) cos(h) + cos(x) sin(h)]� sin(x)
h
= limh!0
sin(x)[cos(h)� 1] + cos(x) sin(h)
h
= limh!0
sin(x)[cos(h)� 1]
h
+ limh!0
cos(x) sin(h)
h
= sin(x) · limh!0
cos(h)� 1
h
+ cos(x) · limh!0
sin(h)
h
= sin(x)(0) + cos(x)(1) = cos(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of sin(x)
To find the derivative of f (x) = sin(x), we now use the definition:
f
0(x) = limh!0
sin(x + h)� sin(x)
h
= limh!0
[sin(x) cos(h) + cos(x) sin(h)]� sin(x)
h
= limh!0
sin(x)[cos(h)� 1] + cos(x) sin(h)
h
= limh!0
sin(x)[cos(h)� 1]
h
+ limh!0
cos(x) sin(h)
h
= sin(x) · limh!0
cos(h)� 1
h
+ cos(x) · limh!0
sin(h)
h
= sin(x)(0) + cos(x)(1) = cos(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of sin(x)
To find the derivative of f (x) = sin(x), we now use the definition:
f
0(x) = limh!0
sin(x + h)� sin(x)
h
= limh!0
[sin(x) cos(h) + cos(x) sin(h)]� sin(x)
h
= limh!0
sin(x)[cos(h)� 1] + cos(x) sin(h)
h
= limh!0
sin(x)[cos(h)� 1]
h
+ limh!0
cos(x) sin(h)
h
= sin(x) · limh!0
cos(h)� 1
h
+ cos(x) · limh!0
sin(h)
h
= sin(x)(0) + cos(x)(1) = cos(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of sin(x)
To find the derivative of f (x) = sin(x), we now use the definition:
f
0(x) = limh!0
sin(x + h)� sin(x)
h
= limh!0
[sin(x) cos(h) + cos(x) sin(h)]� sin(x)
h
= limh!0
sin(x)[cos(h)� 1] + cos(x) sin(h)
h
= limh!0
sin(x)[cos(h)� 1]
h
+ limh!0
cos(x) sin(h)
h
= sin(x) · limh!0
cos(h)� 1
h
+ cos(x) · limh!0
sin(h)
h
= sin(x)(0) + cos(x)(1) = cos(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of cos(x)
The derivative of f (x) = cos(x), is found in a similar way:
f
0(x) = limh!0
cos(x + h)� cos(x)
h
= limh!0
[cos(x) cos(h)� sin(x) sin(h)]� cos(x)
h
= limh!0
cos(x)[cos(h)� 1]� sin(x) sin(h)
h
= limh!0
cos(x)[cos(h)� 1]
h
� limh!0
sin(x) sin(h)
h
= cos(x) · limh!0
cos(h)� 1
h
� sin(x) · limh!0
sin(h)
h
= cos(x)(0)� sin(x)(1) = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of cos(x)
The derivative of f (x) = cos(x), is found in a similar way:
f
0(x) = limh!0
cos(x + h)� cos(x)
h
= limh!0
[cos(x) cos(h)� sin(x) sin(h)]� cos(x)
h
= limh!0
cos(x)[cos(h)� 1]� sin(x) sin(h)
h
= limh!0
cos(x)[cos(h)� 1]
h
� limh!0
sin(x) sin(h)
h
= cos(x) · limh!0
cos(h)� 1
h
� sin(x) · limh!0
sin(h)
h
= cos(x)(0)� sin(x)(1) = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of cos(x)
The derivative of f (x) = cos(x), is found in a similar way:
f
0(x) = limh!0
cos(x + h)� cos(x)
h
= limh!0
[cos(x) cos(h)� sin(x) sin(h)]� cos(x)
h
= limh!0
cos(x)[cos(h)� 1]� sin(x) sin(h)
h
= limh!0
cos(x)[cos(h)� 1]
h
� limh!0
sin(x) sin(h)
h
= cos(x) · limh!0
cos(h)� 1
h
� sin(x) · limh!0
sin(h)
h
= cos(x)(0)� sin(x)(1) = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of cos(x)
The derivative of f (x) = cos(x), is found in a similar way:
f
0(x) = limh!0
cos(x + h)� cos(x)
h
= limh!0
[cos(x) cos(h)� sin(x) sin(h)]� cos(x)
h
= limh!0
cos(x)[cos(h)� 1]� sin(x) sin(h)
h
= limh!0
cos(x)[cos(h)� 1]
h
� limh!0
sin(x) sin(h)
h
= cos(x) · limh!0
cos(h)� 1
h
� sin(x) · limh!0
sin(h)
h
= cos(x)(0)� sin(x)(1) = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of cos(x)
The derivative of f (x) = cos(x), is found in a similar way:
f
0(x) = limh!0
cos(x + h)� cos(x)
h
= limh!0
[cos(x) cos(h)� sin(x) sin(h)]� cos(x)
h
= limh!0
cos(x)[cos(h)� 1]� sin(x) sin(h)
h
= limh!0
cos(x)[cos(h)� 1]
h
� limh!0
sin(x) sin(h)
h
= cos(x) · limh!0
cos(h)� 1
h
� sin(x) · limh!0
sin(h)
h
= cos(x)(0)� sin(x)(1) = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Derivative of cos(x)
The derivative of f (x) = cos(x), is found in a similar way:
f
0(x) = limh!0
cos(x + h)� cos(x)
h
= limh!0
[cos(x) cos(h)� sin(x) sin(h)]� cos(x)
h
= limh!0
cos(x)[cos(h)� 1]� sin(x) sin(h)
h
= limh!0
cos(x)[cos(h)� 1]
h
� limh!0
sin(x) sin(h)
h
= cos(x) · limh!0
cos(h)� 1
h
� sin(x) · limh!0
sin(h)
h
= cos(x)(0)� sin(x)(1) = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
The Results
We now know the derivative functions for the sine and cosine at allx values!
d
dx
[sin(x)] = cos(x)
d
dx
[cos(x)] = � sin(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
As Slopes of Tangents
Compare slopes at points on the graph of f (x) = cos(x) to valueson the graph of f 0(x) = � sin(x):
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
EXAMPLE 1: Finddf
dx
if f (x) = x
2 sin(x) cos(x).
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
EXAMPLE 2a: (Simple Harmonic Motion)
A weight hanging from a spring is stretcheddown 5 units beyond its rest position andreleased at time t = 0. Its position at any latertime t is given by
s(t) = 5 cos(t)
What is the velocity of the weight at time t?
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
EXAMPLE 2b: (Simple Harmonic Motion)
A weight hanging from a spring is stretcheddown 5 units beyond its rest position andreleased at time t = 0. Its position at any latertime t is given by
s(t) = 5 cos(t)
What is the acceleration of the weight at timet?
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
Quiz Yourself
Use the Quotient Rule to find the derivative of the followingfunction:
f (x) =sin(x)
cos(x)
A) �cos(x)
sin(x)
B) � cot(x)
C)cos2(x) + sin2(x)
cos2(x)
D) 1 + tan2(x)
E) sec2(x)
MATH 165 Section 3.5
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Part 1 The Sine and Cosine
The End
MATH 165 Section 3.5
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MATH 165 44–49 Warm-Up Question – Sep. 17, 2018
Use linear approximation
(linearization) to estimate
the increase in volume of a
cube as the side length
changes from 5 ft to 5.1 ft.
A) 0.001 ft
3
B) 0.1 ft
3
C) 7.5 ft
3
D) 7.651 ft
3
E) 125 ft
3
F) 132.651 ft
3
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MATH 165 44–49 Warm-Up Question – Sep. 17, 2018
Use linear approximation
(linearization) to estimate
the increase in volume of a
cube as the side length
changes from 5 ft to 5.1 ft.
A) 0.001 ft
3
B) 0.1 ft
3
*C) 7.5 ft
3
D) 7.651 ft
3
E) 125 ft
3
F) 132.651 ft
3