Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Analysis of Functions of a Single Variable » The Trigonometric and Hyperbolic Functions

Navigation

Table of Contents

Recently Viewed

This feature requires Javascript to be enabled.
 

The Trigonometric and Hyperbolic Functions

Module by: Lawrence Baggett. E-mail the author

Summary: Two theorems covering differentiation 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 zz and w.w.

sin ( z + w ) = sin ( z ) cos ( w ) + cos ( z ) sin ( w ) . sin ( z + w ) = sin ( z ) cos ( w ) + cos ( z ) sin ( w ) .
(1)
cos ( z + w ) = cos ( z ) cos ( w ) - sin ( z ) sin ( w ) . cos ( z + w ) = cos ( z ) cos ( w ) - sin ( z ) sin ( w ) .
(2)
sinh ( z + w ) = sinh ( z ) cosh ( w ) + cosh ( z ) sinh ( w ) . sinh ( z + w ) = sinh ( z ) cosh ( w ) + cosh ( z ) sinh ( w ) .
(3)
cosh ( z + w ) = cosh ( z ) cosh ( w ) + sinh ( z ) sinh ( w ) . cosh ( z + w ) = cosh ( z ) cosh ( w ) + sinh ( z ) sinh ( w ) .
(4)

Proof

We derive the first formula and leave the others to an exercise.

First, for any two real numbers xx and y,y, we have

cos ( x + y ) + i sin ( x + y ) = e i ( x + y ) = e i x e i y = ( cos x + i sin x ) × ( cos y + i sin y ) = cos x cos y - sin x sin y + i ( cos x sin y + sin x cos y ) , cos ( x + y ) + i sin ( x + y ) = e i ( x + y ) = e i x e i y = ( cos x + i sin x ) × ( cos y + i sin y ) = cos x cos y - sin x sin y + i ( cos x sin y + sin x cos y ) ,
(5)

which, equating real and imaginary parts, gives that

cos ( x + y ) = cos x cos y - sin x sin y cos ( x + y ) = cos x cos y - sin x sin y
(6)

and

sin ( x + y ) = sin x cos y + cos x sin y . sin ( x + y ) = sin x cos y + cos x sin y .
(7)

The second of these equations is exactly what we want, but this calculation only shows that it holds for real numbers xx and y.y. We can use the Identity Theorem to show that in fact this formula holds for all complex numbers zz and w.w. Thus, fix a real number y.y. Let f(z)=sinzcosy+coszsiny,f(z)=sinzcosy+coszsiny, and let

g ( z ) = sin ( z + y ) = 1 2 i ( e i ( z + y ) - e - i ( z + y ) = 1 2 i ( e i z e i y - e - i z e - i y ) . g ( z ) = sin ( z + y ) = 1 2 i ( e i ( z + y ) - e - i ( z + y ) = 1 2 i ( e i z e i y - e - i z e - i y ) .
(8)

Then both ff and gg are power series functions of the variable z.z. Furthermore, by the previous calculation, f(1/k)=g(1/k)f(1/k)=g(1/k) for all positive integers k.k. Hence, by the Identity Theorem, f(z)=g(z)f(z)=g(z) for all complex z.z. Hence we have the formula we want for all complex numbers zz and all real numbers y.y.

To finish the proof, we do the same trick one more time. Fix a complex number z.z. Let f(w)=sinzcosw+coszsinw,f(w)=sinzcosw+coszsinw, and let

g ( w ) = sin ( z + w ) = 1 2 i ( e i ( z + w ) - e - i ( z + w ) = 1 2 i ( e i z e i w - e - i z e - i w ) . g ( w ) = sin ( z + w ) = 1 2 i ( e i ( z + w ) - e - i ( z + w ) = 1 2 i ( e i z e i w - e - i z e - i w ) .
(9)

Again, both ff and gg are power series functions of the variable w,w, and they agree on the sequence {1/k}.{1/k}. Hence they agree everywhere, and this completes the proof of the first addition formula.

Exercise 1

  1. Derive the remaining three addition formulas of the preceding theorem.
  2. From the addition formulas, derive the two “half angle” formulas for the trigonometric functions:
    sin2(z)=1-cos(2z)2,sin2(z)=1-cos(2z)2,
    (10)
    and
    cos2(z)=1+cos(2z)2.cos2(z)=1+cos(2z)2.
    (11)

Theorem 2

The trigonometric functions sinsin and coscos are periodic with period 2π;2π; i.e., sin(z+2π)=sin(z)sin(z+2π)=sin(z) and cos(z+2π)=cos(z)cos(z+2π)=cos(z) for all complex numbers z.z.

Proof

We have from the preceding exercise that sin(z+2π)=sin(z)cos(2π)+cos(z)sin(2π),sin(z+2π)=sin(z)cos(2π)+cos(z)sin(2π), so that the periodicity assertion for the sine function will follow if we show that cos(2π)=1cos(2π)=1 and sin(2π)=0.sin(2π)=0. From part (b) of the preceding exercise, we have that

0 = sin 2 ( π ) = 1 - cos ( 2 π ) 2 0 = sin 2 ( π ) = 1 - cos ( 2 π ) 2
(12)

which shows that cos(2π)=1.cos(2π)=1. Since cos2+sin2=1,cos2+sin2=1, it then follows that sin(2π)=0.sin(2π)=0.

The periodicity of the cosine function is proved similarly.

Exercise 2

  1. Prove that the hyperbolic functions sinhsinh and coshcosh are periodic. What is the period?
  2. Prove that the hyperbolic cosine cosh(x)cosh(x) is never 0 for xx a real number, that the hyperbolic tangent tanh(x)=sinh(x)/cosh(x)tanh(x)=sinh(x)/cosh(x) is bounded and increasing from RR onto (-1,1),(-1,1), and that the inverse hyperbolic tangent has derivative given by tanh-1'(y)=1/(1-y2).tanh-1'(y)=1/(1-y2).
  3. Verify that for all y(-1,1)y(-1,1)
    tanh-1(y)=ln(1+y1-y).tanh-1(y)=ln(1+y1-y).
    (13)

Exercise 3: Polar coordinates

Let zz be a nonzero complex number. Prove that there exists a unique real number 0θ<2π0θ<2π such that z=reiθ,z=reiθ, where r=|z|.r=|z|.

HINT: If z=a+bi,z=a+bi, then z=r(ar+bri.z=r(ar+bri. Observe that -1ar1,-1ar1,-1br1,-1br1, and (ar)2+(br)2=1.(ar)2+(br)2=1. Show that there exists a unique 0θ<2π0θ<2π such that ar=cosθar=cosθ and br=sinθ.br=sinθ.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks