Download derivatives of trigonometric functions

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
KEITH CONRAD
There are six basic trigonometric functions:
sin x
cos x
1
1
sin x, cos x, tan x =
, cot x =
, sec x =
, csc x =
.
cos x
sin x
cos x
sin x
We will first compute the derivatives of sin x and cos x. The derivatives of all the other
trigonometric functions will follow from these two using the quotient rule.
Below are the graphs of sin x and cos x.
y
y
y = cos x
y = sin x
x
x
First we find the derivatives of sin x and cos x at x = 0:
sin h
cos h − 1
(1)
sin0 (0) = lim
, cos0 (0) = lim
.
h→0 h
h→0
h
The graph of cos x has a horizontal tangent line at x = 0, so cos0 (0) = 0. Determining
sin0 (0) is more subtle. From the graph of sin x we can see that sin0 (0) > 0. Using radian
measure for angles we will show sin0 (0) in fact is 1.
sin h
Claim. lim
= 1.
h→0 h
Proof of Claim: Pick a small h > 0. In the figure below we mark off an angle h with
respect to the positive x-axis. Let L be the length of a circular arc having radius 1 and
angle h, and M be the length of a circular arc starting at (cos h, 0) with angle h.
y
(cos h, sin h)
sin h
h
L
M
(cos h, 0)
1
x
We will make a comparison of arc lengths. The circular arc with length L is clearly longer
than the vertical segment with length sin h, and this vertical segment in turn is longer than
the circular arc with with length M : M < sin h < L. What are L and M ?
1
2
KEITH CONRAD
When an angle θ is measured in radians, the length of a circular arc it cuts out along a
circle of radius r is rθ. The arc with length L is on a circle with radius 1 and the arc with
length M is on a circle with radius cos h, so if we measure h in radians we get L = 1 · h = h
and M = (cos h)h. Thus (cos h)h < sin h < h, and if we divide by h we get
sin h
< 1.
h
This was derived for small h > 0. Since cos h and (sin h)/h are even functions, the inequality
(2) is true for small h < 0 too, and hence it’s true for all small h 6= 0.
As h → 0 we have cos h → cos 0 = 1 by continuity. Thus (2) tells us (sin h)/h → 1 as
h → 0 by the squeeze theorem. This proves sin0 (0) = 1 when we angles are in radians.
To find the derivatives of sin x and cos x in general we will use their addition formulas:
(2)
cos h <
sin(a + b) = sin a cos b + cos a sin b,
cos(a + b) = cos a cos b − sin a sin b.
The ratio in the limit definition of the derivative of sin x is
(sin x cos h + cos x sin h) − sin x
sin(x + h) − sin x
=
h
h
cos h − 1
sin h
= sin x
+ cos x
.
h
h
Letting h → 0 and looking back at (1) tells us
sin0 (x) =
=
=
=
=
sin(x + h) − sin x
h→0
h
sin h
cos h − 1
+ cos x lim
sin x lim
h→0
h→0
h
h
(sin x)(cos0 (0)) + (cos x)(sin0 (0))
(sin x) · 0 + (cos x) · 1
cos x.
lim
Turning to the derivative of cos x,
cos0 (x) =
=
=
=
=
=
cos(x + h) − cos x
h→0
h
(cos x cos h − sin x sin h) − cos x
lim
h→0
h
cos h − 1
sin h
− sin x
lim cos x
h→0
h
h
cos h − 1
sin h
cos x lim
− sin x lim
h→0
h→0 h
h
0
(cos x)(cos (0)) − (sin x)(sin0 (0))
− sin x.
lim
Using the quotient rule we get the derivative of tan x = sin x/ cos x:
tan0 x =
sin0 x cos x − cos0 x sin x
cos x cos x − (− sin x) sin x
cos2 x + sin2 x
=
=
.
cos2 x
cos2 x
cos2 x
Since sin2 x+cos2 x = 1 the formula simplifies to tan0 x = 1/ cos2 x = sec2 x. The derivatives
of the other trigonometric functions can be found in a similar way:
cos x 0 (cos0 x)(sin x) − (cos x) sin0 x
− cos2 x − sin2 x
1
cot0 x =
=
=
= − 2 = − csc2 x
2
2
sin x
sin x
sin x
sin x
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
and
0
sec x =
1
cos x
0
=
3
− cos0 x
sin x
=
= sec x tan x
2
cos x
cos2 x
and
0
cos x
1
− sin0 x
= − 2 = − csc x cot x.
csc x =
=
2
sin x
sin x
sin x
Remark. It is critical that we measure angles in radians for the above derivative formulas
to be correct. If we measure angles in degrees then the formulas are wrong. Compare the
graphs of sin x and cos x below, all drawn to the same scale, when x measured first in
radians and then in degrees. For example, sin0 (0) is nearly 0 when we measure angles in
degrees. Precisely, when angles are in degrees sin0 (0) = π/180 ≈ .0174.
0
y
y = sin x
x in radians
x
y
y = sin x
x in degrees
x
y
y = cos x
x in radians
x
y
y = cos x
x in degrees
x