The derivative of sine is cosine. This fact alone tells how much more complicated trigonometric functions are than regular polynomials.

1. Section 3.4
Derivatives of Trigonometric Functions
Math 1a
February 25, 2008
Announcements
Get 50% of all ALEKS points between now and 3/7
Problem Sessions Sunday, Thursday, 7pm, SC 310
Office hours Tuesday, Wednesday 2–4pm SC 323
Midterm I Friday 2/29 in class (up to §3.2)

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

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
cos θ 1 θ
As θ → 0, the left and right
sides tend to 1. So, then,
must the middle
expression.

4. Now
1 − cos θ 1 − cos θ 1 + cos θ 1 − cos2 θ
= · =
θ θ 1 + cos θ θ(1 + cos θ)
2
sin θ sin θ cos θ
= = ·
θ(1 + cos θ) θ 1 + cos θ
So
1 − cos θ sin θ cos θ
lim = lim · lim
θ→0 θ θ→0 θ θ→0 1 + cos θ
= 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:
d sin(x + h) − sin x
sin x = lim
dx h→0 h

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

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

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

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

11. Illustration of Sine and Cosine
y
x
π π 0 π π
− 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
Fill in the table:
y y
sin x cos x
cos x − sin x
tan x
cot x
sec x
csc x

14. Derivatives of tangent and secant
Fill in the table:
y y
sin x cos x
cos x − sin x
tan x sec2 x
cot x
sec x
csc x

15. Derivatives of tangent and secant
Fill in the table:
y y
sin x cos x
cos x − sin x
tan x sec2 x
cot x − csc2 x
sec x
csc x

16. Derivatives of tangent and secant
Fill in the table:
y y
sin x cos x
cos x − sin x
tan x sec2 x
cot x − csc2 x
sec x sec x tan x
csc x

17. Derivatives of tangent and secant
Fill in the table:
y y
sin x cos x
cos x − sin x
tan x sec2 x
cot x − csc2 x
sec x sec x tan x
csc x − csc x cot x

18. Example
Find
sin 2θ
lim
θ→0 θ

19. Example
Find
sin 2θ
lim
θ→0 θ
Solution (i)
Use a trig identity:
sin 2θ 2 sin θ cos θ sin θ
lim = lim = 2 · lim · lim cos θ = 2 · 1 · 1 = 1
θ→0 θ θ→0 θ θ→0 θ θ→0

20. Example
Find
sin 2θ
lim
θ→0 θ
Solution (i)
Use a trig identity:
sin 2θ 2 sin θ cos θ sin θ
lim = lim = 2 · lim · lim cos θ = 2 · 1 · 1 = 1
θ→0 θ θ→0 θ θ→0 θ θ→0
Solution (ii)
Use a change of variable. Let ϕ = 2θ. Then
sin 2θ sin 2θ sin ϕ
lim = 2 lim = 2 lim =1
θ→0 θ θ→0 2θ ϕ→0 ϕ