This document contains notes from a calculus class that discusses: 1) Two important limits involving trigonometric functions - the limit of sin(θ)/θ as θ approaches 0 equals 1, and the limit of (cos(θ) - 1)/θ as θ approaches 0 equals 0. 2) The derivatives of sine and cosine - the derivative of sine is cosine, and the derivative of cosine is the negative of sine. 3) The derivative of tangent is secant squared, and the derivative of secant is secant times tangent.

1. Sections 3.4
Derivatives of Trigonometric Functions
Math S-1ab
Calculus I and II
October 26, 2007
Announcements
Midterm is done. Average=84.2, Median=85

2. Two important trigonometric limits
Theorem
The following two limits hold:
sin θ
lim =1
θ→0 θ
cos θ − 1
lim =0
θ
θ→0

3. Proof of the Sine Limit
Proof.
Notice
sin θ ≤ θ ≤ tan θ
Divide by sin θ:
θ 1
1≤ ≤
sin θ cos θ
Take reciprocals:
sin θ θ tan θ
θ sin θ
1≥ ≥ cos θ
θ
cos θ 1
As θ → 0, the left and right
sides tend to 1. So, then,
must the middle
expression.

4. Now
1 − cos2 θ
1 − cos θ 1 − cos θ 1 + cos θ
·
= =
θ θ 1 + cos θ θ(1 + cos θ)
2
sin θ sin θ θ
·
= =
θ(1 + cos θ) θ 1 + cos θ
So
1 − cos θ sin θ θ
·
lim = lim lim
θ θ→0 θ θ→0 1 + cos θ
θ→0
= 1 · 0 = 0.

5. Derivatives of Sine and Cosine
Theorem
d
sin x = cos x.
dx
Proof.
From the definition:

6. Derivatives of Sine and Cosine
Theorem
d
sin x = cos x.
dx
Proof.
From the definition:
sin(x + h) − sin x
d
sin x = lim
dx h
h→0

7. Derivatives of Sine and Cosine
Theorem
d
sin x = cos x.
dx
Proof.
From the definition:
sin(x + h) − sin x
d
sin x = lim
dx h
h→0
(sin x cos h + cos x sin h) − sin x
= lim
h
h→0

8. Derivatives of Sine and Cosine
Theorem
d
sin x = cos x.
dx
Proof.
From the definition:
sin(x + h) − sin x
d
sin x = lim
dx h
h→0
(sin x cos h + cos x sin h) − sin x
= lim
h
h→0
cos h − 1 sin h
= sin x · lim + cos x · lim
h h→0 h
h→0

9. Derivatives of Sine and Cosine
Theorem
d
sin x = cos x.
dx
Proof.
From the definition:
sin(x + h) − sin x
d
sin x = lim
dx h
h→0
(sin x cos h + cos x sin h) − sin x
= lim
h
h→0
cos h − 1 sin h
= sin x · lim + cos x · lim
h h→0 h
h→0
= sin x · 0 + cos x · 1 = cos x

10. Illustration of Sine and Cosine
y
x
π π π π
0
−
2 2 sin x

11. Illustration of Sine and Cosine
y
x
π π π π
0
− cos x
2 2 sin x

12. Derivatives of Sine and Cosine
Theorem
d
sin x = cos x.
dx
d
cos x = − sin x.
dx

13. Derivatives of tangent and secant
Example
d
Find tan x
dx

14. Derivatives of tangent and secant
Example
d
Find tan x
dx
Answer
sec2 x.

15. Derivatives of tangent and secant
Example
d
Find tan x
dx
Answer
sec2 x.
Example
d
Find sec x
dx

16. Derivatives of tangent and secant
Example
d
Find tan x
dx
Answer
sec2 x.
Example
d
Find sec x
dx
Answer
sec x tan x.