We compute the derivatives of sine, cosine, tangent, cotangent, secant, cosecant

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.