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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Part 1 The Sine and Cosine

EXAMPLE 1: Finddf

dx

if f (x) = x

2 sin(x) cos(x).

MATH 165 Section 3.5

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

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

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

Part 1 The Sine and Cosine

The End

MATH 165 Section 3.5

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

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