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3.3: Derivatives of Trigonometric Functions
Telyn Kusalik
3.3.1 Derivatives of sin and cos
Objective: Find the derivative of
f (x) = 2 sin x + 3 cos x.
MATH 111 Fall 2015
Derivatives of sin and cos graphically
Let’s look at the graphs of y = sin x and y = cos x and attempt
to take the derivatives of these functions graphically.
The graph of the derivative of y = sin x looks an awful lot like
the graph of y = cos x.
The graph of the derivative of y = cos x looks like an
upside-down sin x.
These are not coincidences. The derivatives of sin x and cos x
are as follows.
(sin x)0 = cos x
(cos x)0 = − sin x
Radian Measure and Calculus
The fact that the derivative of sin x is cos x is a fact that
crucially depends on us using radian measure. If the sin
function is graphed using radian measure, the slope when
x = 0 is exactly 1, and cos(0) = 1.
However, if we graph sin x using degree measure, we notice
that the slope at x = 0 is a lot smaller than 1. It turns out to be
π
exactly 180
. Using degree measure, the rule for the derivative of
sin x would be:
π
(sin x)0 =
cos x
180
So the reason we use radian measure is that it makes the
calculus of trigonometric functions much simpler.
Derivations of Derivative Rules for sin and cos
Derivatives Involving sin x and cos x
In order to derive the derivative rules for sin and cos, we need
the two limits you found on your first assignment:
lim
x→0
sin x
=1
x
lim
x→0
cos x − 1
=0
x
We also need the sum identities for sin and cos:
sin(a + b) = sin a cos b + cos a sin b
cos(a + b) = cos a cos b − sin a sin b
Asking you to find the derivative of sin x or cos x on its own is
too easy a problem for me to ask. Usually, I’ll ask problems
involving the product or quotient rule as well as a sin or cos:
Example 1: Find the derivative of f (x) =
Example 2: Differentiate y =
√
x sin x.
cos x
1−sin x
Using these tools, we can use the definition of the derivative to
find the derivatives of sin and cos. I will derive the derivative
d
formula dx
(sin x) = cos x as an example.
Higher Derivatives of sin and cos
Tangent Line Problems
One of the interesting things about sin and cos functions is that
when we take higher and higher derivatives of them, we end up
getting back where we started. Let’s start with y = cos x
y 0 = − sin x
y 00 = − cos x
y 000 = sin x
y (4) = cos x
y (5) = − sin x y (6) = − cos x y (7) = sin x y (8) = cos x
In general, any derivative of order which is a multiple of 4 will
be cos x. This makes it easy to find very high derivatives like:
d 43
cos x.
dx 43
The 40th derivative is cos x, so the 41st is − sin x, so the 42nd is
− cos x, so the 43rd is sin x.
Just like with other types of functions, tangent line problems
can be written than involve trigonometric functions. The
difficulty many students have is not with the actual calculus of
these functions, but in applying pre-calculus skills such as
solving trigonometric equations.
Example 3 : For what values of x does the graph of f have a
horizontal tangent?
f (x) = ex cos x
Derivatives of tan, cot, sec and csc using sin and cos
3.3.2 Derivatives of other Trigonometric Functions
Objective: Find the derivative of
f (x) = 2 sec x + 3 tan x.
Since the trigonometric functions tan, cot, sec, and csc can all
be expressed using sin and cos, their derivatives can be taken
using the quotient rule:
0
(tan x) =
sin x
cos x
0
=
cos2 x + sin2 x
= sec2 x
cos2 x
I could ask you, on a quiz or test, to “prove” the derivative rule
for one of these trigonometric functions:
d
Example 4: Prove that dx
(sec x) = sec x tan x.
How to Remember Trigonometric Derivatives
Most students usually choose to memorize the derivatives of
tan, cot, sec and csc rather than to go through the derivations
like on the previous slide. One thing to keep in mind to help
remember them is that the derivatives of the “co” functions are
basically the same as the derivatives of the “original” functions
except they are “co” and negative:
0
0
(sin x) = cos x
(cos x) = − sin x
(tan x)0 = sec2 x
(cot x)0 = − csc2 x
(sec x)0 = sec x tan x
(csc x)0 = − csc x cot x
Derivatives Involving tan, cot, sec and csc
Again, just like with sin and cos, most derivative problems in
this module will involve the product and quotient rule as well as
the trigonometric functions.
Example 5: Differentiate g(θ) = eθ (tan θ − θ).
Example 6: Find the derivative of y =
1−sec x
tan x .
More Complicated Derivatives
The most complicated derivatives I can ask for at this point are
those that involve more than one product or quotient rule:
Example 7: Differentiate y = x 2 sin x tan x.