Sections 3.4Derivatives of Trigonometric Functions

Math S-1abCalculus I and II

October 26, 2007

Announcements

I Midterm is done. Average=84.2, Median=85

Two important trigonometric limits

TheoremThe following two limits hold:

I limθ→0

sin θ

θ= 1

I limθ→0

cos θ − 1

θ= 0

Proof of the Sine Limit

θsin θ

cos θ

θ tan θ

−1 1

Proof.Notice

sin θ ≤ θ ≤ tan θ

Divide by sin θ:

1 ≤ θ

sin θ≤ 1

cos θ

Take reciprocals:

1 ≥ sin θ

θ≥ cos θ

As θ → 0, the left and rightsides tend to 1. So, then,must the middleexpression.

Now

1− cos θ

θ=

1− cos θ

θ· 1 + cos θ

1 + cos θ=

1− cos2 θ

θ(1 + cos θ)

=sin2 θ

θ(1 + cos θ)=

sin θ

θ· θ

1 + cos θ

So

limθ→0

1− cos θ

θ=

(limθ→0

sin θ

θ

)·(

limθ→0

θ

1 + cos θ

)= 1 · 0 = 0.

Derivatives of Sine and Cosine

Theorem

d

dxsin x = cos x .

Proof.From the definition:

d

dxsin x = lim

h→0

sin(x + h)− sin x

h

= limh→0

(sin x cos h + cos x sin h)− sin x

h

= sin x · limh→0

cos h − 1

h+ cos x · lim

h→0

sin h

h

= sin x · 0 + cos x · 1 = cos x

Derivatives of Sine and Cosine

Theorem

d

dxsin x = cos x .

Proof.From the definition:

d

dxsin x = lim

h→0

sin(x + h)− sin x

h

= limh→0

(sin x cos h + cos x sin h)− sin x

h

= sin x · limh→0

cos h − 1

h+ cos x · lim

h→0

sin h

h

= sin x · 0 + cos x · 1 = cos x

Derivatives of Sine and Cosine

Theorem

d

dxsin x = cos x .

Proof.From the definition:

d

dxsin x = lim

h→0

sin(x + h)− sin x

h

= limh→0

(sin x cos h + cos x sin h)− sin x

h

= sin x · limh→0

cos h − 1

h+ cos x · lim

h→0

sin h

h

= sin x · 0 + cos x · 1 = cos x

Derivatives of Sine and Cosine

Theorem

d

dxsin x = cos x .

Proof.From the definition:

d

dxsin x = lim

h→0

sin(x + h)− sin x

h

= limh→0

(sin x cos h + cos x sin h)− sin x

h

= sin x · limh→0

cos h − 1

h+ cos x · lim

h→0

sin h

h

= sin x · 0 + cos x · 1 = cos x

Derivatives of Sine and Cosine

Theorem

d

dxsin x = cos x .

Proof.From the definition:

d

dxsin x = lim

h→0

sin(x + h)− sin x

h

= limh→0

(sin x cos h + cos x sin h)− sin x

h

= sin x · limh→0

cos h − 1

h+ cos x · lim

h→0

sin h

h

= sin x · 0 + cos x · 1 = cos x

Illustration of Sine and Cosine

x

y

π −π2

0 π

2

π

sin x

cos x

Illustration of Sine and Cosine

x

y

π −π2

0 π

2

π

sin xcos x

Derivatives of Sine and Cosine

Theorem

Id

dxsin x = cos x.

Id

dxcos x = − sin x.

Derivatives of tangent and secant

Example

Findd

dxtan x

Answersec2 x .

Example

Findd

dxsec x

Answersec x tan x .

Derivatives of tangent and secant

Example

Findd

dxtan x

Answersec2 x .

Example

Findd

dxsec x

Answersec x tan x .

Derivatives of tangent and secant

Example

Findd

dxtan x

Answersec2 x .

Example

Findd

dxsec x

Answersec x tan x .

Derivatives of tangent and secant

Example

Findd

dxtan x

Answersec2 x .

Example

Findd

dxsec x

Answersec x tan x .