Differential Calculus 201-NYA-05
Vincent Carrier
Derivative of the Sine and Cosine Functions
1) A Very Important Limit
For t > 0,
y
6
1
sin t
≤
t
≤
tan t
sin t
≤
t
≤
sin t
cos t
1
≤
t
sin t
≤
1
cos t
1
≥
sin t
t
≥
cos t
cos t
≤
sin t
t
≤
1
tan t
1
t
sin t
t
1 x
lim+ cos t ≤
t→0
cos 0
≤
lim+
sin t
≤
t
lim+
sin t
≤
t
1
lim+
sin t
≤
t
1
t→0
t→0
1
≤
t→0
lim 1
t→0+
This implies
lim+
t→0
sin t
= 1.
t
On the other hand,
lim−
t→0
sin(−t)
− sin t
sin t
sin t
= lim+
= lim+
= lim+
= 1.
t→0
t→0
t→0
t
−t
−t
t
Therefore,
sin t
= 1.
t→0 t
lim
2) Addition Formulas for Sine and Cosine
The addition formulas for sine and cosine are
sin(s + t) = sin s cos t + cos s sin t,
cos(s + t) = cos s cos t − sin s sin t.
Proof:
y
6
1
r(cos(s + t), sin(s + t))
t
1
a
d
s
b
t
c
1
x
It can be seen by looking at the above illustration that
a = sin s cos t,
b = cos s sin t,
c = cos s cos t,
d = sin s sin t.
Therefore,
sin(s + t) = a + b
= sin s cos t + cos s sin t,
cos(s + t) = c − d
= cos s cos t − sin s sin t.
3) Derivative of Sine:
d
sin x = cos x.
dx
Proof:
sin(x + h) − sin x
d
sin x = lim
h→0
dx
h
sin x cos h + cos x sin h − sin x
h→0
h
= lim
sin x(cos h − 1) + cos x sin h
h→0
h
sin x(cos h − 1) cos x sin h
+
= lim
h→0
h
h
= lim
sin x(cos h − 1)
cos x sin h
+ lim
h→0
h→0
h
h
cos h − 1 cos h + 1
sin h
= sin x lim
+ cos x lim
h→0
h→0 h
h
cos h + 1
= lim
cos2 h − 1
+ cos x
h→0 h(cos h + 1)
= sin x lim
− sin2 h
+ cos x
h→0 h(cos h + 1)
= sin x lim
= − sin x lim
h→0
sin h
cos h + 1
sin h
= − sin x lim
h→0 cos h + 1
= − sin x
= − sin x
= cos x.
sin 0
cos 0 + 1
0
1+1
sin h
h
sin h
lim
h→0 h
(1) + cos x
(1) + cos x
+ cos x
+ cos x
4) Derivative of Cosine:
d
cos x = − sin x.
dx
Proof:
sin2 x + cos2 x = 1
2 sin x cos x + 2 cos x
cos x
d
cos x
dx
d
cos x
dx
= 0
= − sin x cos x
d
cos x = − sin x.
dx
Graph of sin x:
y
6
1
−2π
3π
−
2
−π
0
π
−
2
π
2
π
3π
2
2π
π
2
π
3π
2
2π
x
−1
Graph of cos x:
y
6
1
−2π
3π
−
2
−π
0
π
−
2
−1
x