Derivatives of sine and cosine
Differentiation Rules for sine and cosine
- sin'(x) = cos(x) and
- cos'(x) = -sin(x).
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The following sections explain why the derivative of cos(x) is -sin(x) and
why the derivative of sin(x) is cos(x).
From Earlier Pages, recal
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lim
h® 0 |
cos(h) - 1
h |
= |
0 |
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lim
h® 0 |
sin(h)
h |
= |
1 |
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6 A. Derivatives of sin(x) and cos(x) - real number way
A complex number way follows below.
Recall the angle sum formulas
cos(a+b) = cos(a)cos(b)
- sin(a)sin(b)
sin(a+b) = sin(a)cos(b)
+ cos(a)sin(b)
Now
cos(x+h) = cos(x)cos(h)- sin(x)sin(h)
gives
cos(x+h) -cos(x) = cos(x)cos(h)- sin(x)sin(h) - cos(x)
= cos(x)cos(h) - cos(x) - sin(x)sin(h)
= cos(h)cos(x) - cos(x) - sin(x)sin(h)
= (cos(h)-1) cos(x) - sin(x)sin(h)
Now
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lim
h® 0 |
cos(x+h) -cos(x)
h |
= |
lim
h® 0 |
(cos(h)-1)cos(x)- sin(x)sin(h)
h |
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= |
lim
h® 0 |
(cos(h)-1)
h |
cos(x) - sin(x) |
sin(h)
h |
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= |
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0 |
cos(x) - sin(x) |
1 |
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= |
- |
sin(x) |
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Therefore cos'(x) = -sin(x).
Similarly, sin(x+h) = sin(x)cos(h) + cos(x)sin(h)
gives
sin(x+h) - sin(x) = sin(x)cos(h) + cos(x)sin(h)
- sin(x)
= (cos(h)- 1) sin(x) + cos(x) sin(x)
Now
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lim
h ® 0 |
sin(x+h) -sin(x)
h |
= |
lim
h® 0 |
(cos(h)-1)sin(x)+
cos(x)sin(h)
h |
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= |
lim
h® 0 ) |
(cos(h)-1)
h |
sin(x) +
cos(x) |
sin(h)
h |
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= |
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0 |
sin(x) +
cos(x) |
1 |
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= |
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cos(x) |
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Therefore sin'(x) = cos(x).
Derivatives of Basic Trig Functions
Now the derivatives of the other four basic trig functions tan(x),
cot(x), sec(x) and csc(x) follow from the quotient and reciprocals rules for
differentiation along with some simplification based on the definition of trig
functions in terms of sines and cosines, and the Pythagorean identity.
6B. Derivatives of sin(x) and cos(x) - complex number way (optional)
| If you knew about complex
numbers and
cis(x) = cos(x) + i sin(x)
we could at this point calculate the derivative of cis(x),
and its real and imaginary parts cos(x) and sin(x) as follows.
Note if f(x) = h(x) + ig(x) for a pair of real
valued function of pair of real-valued variable x, we take
lim x® a f(x) = lim x® a h(x) + i lim x® a g(x)
These complex-valued limits inherit many of the
properties of the real-valued limits studied earlier. Proofs are
omitted. It is left to the student (or another course) to formally state
the properties.
It is easy to show lim x® a f(x) k = [lim x® a f(x) ] k for any complex number k that is independent of
x.
By our definition of complex-valued limits (saying how to
compute them, defines them), we have
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lim h® 0 |
cis(h)- 1
h |
= |
lim h® 0 |
(cos(h) -1) + i sin(h)
h |
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= |
lim h® 0 |
(cos(h) -1)
h |
+ i |
sin(h)
h |
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= |
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0 |
+ i |
1 |
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= |
i |
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Now
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lim h® 0 |
cis(x+h)- cis(x)
h |
= |
lim h® 0 |
(cis(h)- 1)cis(x)
h |
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= |
i |
cis(x) |
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That is to say
cis'(x) = i cis(x) = i [cos(x) + i sin(x)] = -
sin(x) + i sin(x).
But cis'(x) = cos'(x) + i sin'(x) as well. The argument
that cis'(x) implies cos'(x) and sin'(x) exist as well, and both are real.
So
cis'(x) = cos'(x) + i sin'(x) = - sin(x) + i sin(x).
and comparison of real and imaginary (equating them) gives
cos'(x) = -sin(x) and sin'(x) = cos(x) |
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Calculus Appetizers
& Lessons
Starter Guide (Views)
Real Player Videos
3. Derivative Motivation
3. Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x) (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3. Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
YOU are better than YOU think. Show yourself how:
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