Download Differentiation, Local Behavior: The Trigonometric

OpenStax-CNX module: m36196
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Differentiation, Local Behavior:
The Trigonometric and Hyperbolic
Functions
∗
Lawrence Baggett
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
Two theorems covering dierentiation of trigonometric and hyperbolic functions, including practice
exercises corresponding to the theorems.
The laws of exponents and the algebraic connections between the exponential function and the trigonometric and hyperbolic functions, give the following addition formulas:
Theorem 1:
The following identities hold for all complex numbers
z
and
w.
sin (z + w) = sin (z) cos (w) + cos (z) sin (w) .
(1)
cos (z + w) = cos (z) cos (w) − sin (z) sin (w) .
(2)
sinh (z + w) = sinh (z) cosh (w) + cosh (z) sinh (w) .
(3)
cosh (z + w) = cosh (z) cosh (w) + sinh (z) sinh (w) .
(4)
Proof:
We derive the rst formula and leave the others to an exercise.
First, for any two real numbers
cos (x + y) + isin (x + y)
x
and
y,
we have
=
ei(x+y)
=
eix eiy
=
(cosx + isinx) × (cosy + isiny)
=
cosxcosy − sinxsiny + i (cosxsiny + sinxcosy) ,
(5)
which, equating real and imaginary parts, gives that
cos (x + y) = cosxcosy − sinxsiny
∗
†
Version 1.2: Dec 9, 2010 4:35 pm +0000
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(6)
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and
sin (x + y) = sinxcosy + cosxsiny.
(7)
The second of these equations is exactly what we want, but this calculation only shows that it
holds for real numbers
x
and
y. We can use the Identity Theorem to show that in fact this formula
z and w. Thus, x a real number y. Let f (z) = sinzcosy + coszsiny,
holds for all complex numbers
and let
g (z) = sin (z + y) =
Then both
calculation,
g (z)
for all
numbers
1
2i
1 iz iy
e e − e−iz e−iy .(8)
ei(z+y) − e−i(z+y) =
2i
f and g are power series functions of the variable z. Furthermore, by the previous
f (1/k) = g (1/k) for all positive integers k. Hence, by the Identity Theorem, f (z) =
complex z. Hence we have the formula we want for all complex numbers z and all real
y.
To nish the proof, we do the same trick one more time. Fix a complex number
sinzcosw + coszsinw,
g (w) = sin (z + w) =
Again, both
{1/k}.
f
and
g
z.
Let
f (w) =
and let
1
2i
1 iz iw
ei(z+w) − e−i(z+w) =
e e − e−iz e−iw .(9)
2i
are power series functions of the variable
w,
and they agree on the sequence
Hence they agree everywhere, and this completes the proof of the rst addition formula.
Exercise 1
a. Derive the remaining three addition formulas of the preceding theorem.
b. From the addition formulas, derive the two half angle formulas for the trigonometric functions:
sin2 (z) =
1 − cos (2z)
,
2
(10)
cos2 (z) =
1 + cos (2z)
.
2
(11)
and
Theorem 2:
The trigonometric functions
and
cos (z + 2π) = cos (z)
Proof:
sin
and
cos
are periodic with period
We have from the preceding exercise that
2π;
i.e.,
sin (z + 2π) = sin (z)
z.
for all complex numbers
sin (z + 2π) = sin (z) cos (2π) + cos (z) sin (2π) , so
cos (2π) = 1 and
that the periodicity assertion for the sine function will follow if we show that
sin (2π) = 0.
From part (b) of the preceding exercise, we have that
0 = sin2 (π) =
which shows that
cos (2π) = 1.
Since
1 − cos (2π)
2
cos2 + sin2 = 1,
it then follows that
(12)
sin (2π) = 0.
The periodicity of the cosine function is proved similarly.
Exercise 2
a. Prove that the hyperbolic functions
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sinh
and
cosh
are periodic. What is the period?
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cosh (x) is never 0 for x a real number, that the hyperbolic
tanh (x) = sinh (x) /cosh (x) is bounded and increasing from R onto (−1, 1) , and
−1 '
that the inverse hyperbolic tangent has derivative given by tanh
(y) = 1/ 1 − y 2 .
Verify that for all y ∈ (−1, 1)
r
1+y
−1
tanh (y) = ln
.
(13)
1−y
b. Prove that the hyperbolic cosine
tangent
c.
Exercise 3: Polar coordinates
Let
z
be a nonzero complex number. Prove that there exists a unique real number
z = reiθ , where r = |z|.
HINT: If z = a + bi, then z = r
2
a 2
+ rb = 1. Show that there exists
r
0 ≤ θ < 2π
such that
http://cnx.org/content/m36196/1.2/
a
r
+ rb i.
a unique
a
r
a
such that
r
Observe that
0 ≤ θ < 2π
−1 ≤
b
r
b
and
r
≤ 1,−1 ≤
≤ 1,
= cosθ
= sinθ.
and