Download 19. Derivative of sine and cosine

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:
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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).
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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
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This completes the verification of the two trigonometric limit formulas.
19.2. Statement
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Derivative of sine and cosine.
d
(a)
[sin x] = cos x,
dx
d
(b)
[cos x] = − sin x.
dx
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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)
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((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.
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Derivative of sine and cosine
Two trigonometric limits
Statement
Examples
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19.3. Examples
19.3.1
Solution
Example
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Find the derivative of f (x) = 3 cos x + 5 sin x.
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We use the rules of this section after first applying the sum rule and the constant
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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
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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.
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Therefore, the x-coordinates of the desired points are
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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).
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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.
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