19. Derivative of sine and cosine
Derivative of sine and cosine
Two trigonometric limits
19.1. Two trigonometric limits
Statement
Examples
The rules for finding the derivative of the functions sin x and cos x depend on two limits
(that are used elsewhere in calculus as well):
Two trigonometric limits.
(a) lim
θ→0
sin θ
= 1,
θ
cos θ − 1
(b) lim
= 0.
θ→0
θ
The verification we give of the first formula is based on the pictured wedge of the unit
circle:
Table of Contents
JJ
II
J
I
Page 1 of 7
Back
Print Version
Home Page
Derivative of sine and cosine
Two trigonometric limits
Statement
Examples
The segment tagged with sin θ has this length by the definition of sin θ as the y-coordinate
of the point P on the unit circle corresponding to the angle θ. The line segment tagged
with tan θ has this length since looking at the large triangle, we have tan θ = o/a = o.
The arc tagged with θ has this length by the definition of radian measure of an angle. The
diagram reveals the inequalities
sin θ < θ < tan θ.
The first inequality implies (sin θ)/θ < 1; the second says θ < (sin θ)/(cos θ), implying that
cos θ < (sin θ)/θ. Therefore,
sin θ
< 1.
cos θ <
θ
As θ goes to 0, both ends go to 1 forcing the middle expression to go to 1 as well (by the
squeeze theorem). This establishes (a).
Table of Contents
JJ
II
J
I
Page 2 of 7
Back
Print Version
Home Page
For the second formula, we use a method that is similar to our rationalization method, as
well as the main trigonometric identity, and finally the first formula:
cos θ − 1
cos θ − 1 cos θ + 1
= lim
·
θ→0
θ→0
θ
θ
cos θ + 1
cos2 θ − 1
= lim
θ→0 θ(cos θ + 1)
lim
− sin2 θ
θ→0 θ(cos θ + 1)
sin θ
sin θ
= − lim
·
θ→0 θ
cos θ + 1
sin θ
sin θ
· lim
= − lim
θ→0 θ
θ→0 cos θ + 1
= −1 · 0 = 0.
Derivative of sine and cosine
Two trigonometric limits
Statement
Examples
= lim
Table of Contents
JJ
II
J
I
This completes the verification of the two trigonometric limit formulas.
19.2. Statement
Page 3 of 7
Derivative of sine and cosine.
d
(a)
[sin x] = cos x,
dx
d
(b)
[cos x] = − sin x.
dx
Back
Print Version
Home Page
We verify only the first of these derivative formulas. With f (x) = sin x, the formula says
f 0 (x) = cos x:
f (x + h) − f (x)
h→0
h
sin(x + h) − sin x
= lim
h→0
h
(sin x cos h + cos x sin h) − sin x
= lim
h→0
h
cos h − 1
sin h
= lim sin x ·
+ cos x ·
h→0
h
h
cos h − 1
sin h
= sin x · lim
+ cos x · lim
h→0
h→0 h
h
= sin x · 0 + cos x · 1
Derivative of sine and cosine
Two trigonometric limits
Statement
f 0 (x) = lim
Examples
((4) of 4.3)
Table of Contents
((b) and (a))
= cos x.
The formula says that f (x) = sin x has general slope function f 0 (x) = cos x, so the height
of the graph of the cosine function at x should be the slope of the graph of the sine function
at x. The following figures show this relationship for x a multiple of π/2.
JJ
II
J
I
Page 4 of 7
Back
Print Version
Home Page
Derivative of sine and cosine
Two trigonometric limits
Statement
Examples
Table of Contents
19.3. Examples
19.3.1
Solution
Example
JJ
II
J
I
Page 5 of 7
Find the derivative of f (x) = 3 cos x + 5 sin x.
Back
We use the rules of this section after first applying the sum rule and the constant
Print Version
Home Page
Derivative of sine and cosine
multiple rule:
d
[3 cos x + 5 sin x]
dx
d
d
= 3 [cos x] + 5 [sin x]
dx
dx
= 3(− sin x) + 5(cos x)
Two trigonometric limits
f 0 (x) =
Statement
Examples
= −3 sin x + 5 cos x.
19.3.2 Example
line has slope 1/2.
Find all points on the graph of f (x) = sin x at which the tangent
Table of Contents
0
Solution The general slope function for this function is its derivative, which is f (x) =
cos x. We get the x-coordinates of the desired points by solving f 0 (x) = 1/2, that is,
cos x = 1/2. Picturing the unit circle and using the 30-60-90 triangle we find that x = π/3
(60◦ ) in the first quadrant yields a cosine of 1/2, as does x = −π/3 in the fourth quadrant.
A multiple of 2π added to these angles produces the same cosine.
JJ
II
J
I
Therefore, the x-coordinates of the desired points are
Page 6 of 7
π/3 + 2πn,
−π/3 + 2πn
(n any integer).
√
The y-coordinate corresponding to x = π/3 is f (π/3) = sin(π/3) = 3/2 and this is also
the y-coordinate corresponding to x =√π/3 + 2πn for any n. Similarly, the y-coordinate
corresponding to x = −π/3 + 2πn is − 3/2. The desired points are
√
√
(π/3 + 2πn, 3/2), (−π/3 + 2πn, − 3/2)
(n any integer).
Back
Print Version
Home Page
Derivative of sine and cosine
19 – Exercises
Two trigonometric limits
Statement
19 – 1
Find all points on the graph of f (x) = cos x at which the tangent line has slope
√
Examples
3/2.
Table of Contents
JJ
II
J
I
Page 7 of 7
Back
Print Version
Home Page