# 4.5 The trigonometric and hyperbolic functions

 Page 1 / 1
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 andhyperbolic functions, give the following “addition formulas:”

The following identities hold for all complex numbers $z$ and $w.$

$sin\left(z+w\right)=sin\left(z\right)cos\left(w\right)+cos\left(z\right)sin\left(w\right).$
$cos\left(z+w\right)=cos\left(z\right)cos\left(w\right)-sin\left(z\right)sin\left(w\right).$
$sinh\left(z+w\right)=sinh\left(z\right)cosh\left(w\right)+cosh\left(z\right)sinh\left(w\right).$
$cosh\left(z+w\right)=cosh\left(z\right)cosh\left(w\right)+sinh\left(z\right)sinh\left(w\right).$

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

First, for any two real numbers $x$ and $y,$ we have

$\begin{array}{ccc}\hfill cos\left(x+y\right)+isin\left(x+y\right)& =& {e}^{i\left(x+y\right)}\hfill \\ & =& {e}^{ix}{e}^{iy}\hfill \\ & =& \left(cosx+isinx\right)×\left(cosy+isiny\right)\hfill \\ & =& cosxcosy-sinxsiny+i\left(cosxsiny+sinxcosy\right),\hfill \end{array}$

which, equating real and imaginary parts, gives that

$cos\left(x+y\right)=cosxcosy-sinxsiny$

and

$sin\left(x+y\right)=sinxcosy+cosxsiny.$

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 holds for all complex numbers $z$ and $w.$ Thus, fix a real number $y.$ Let $f\left(z\right)=sinzcosy+coszsiny,$ and let

$g\left(z\right)=sin\left(z+y\right)=\frac{1}{2i}\left({e}^{i\left(z+y\right)}-{e}^{-i\left(z+y\right)}=\frac{1}{2i}\left({e}^{iz}{e}^{iy}-{e}^{-iz}{e}^{-iy}\right).$

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

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

$g\left(w\right)=sin\left(z+w\right)=\frac{1}{2i}\left({e}^{i\left(z+w\right)}-{e}^{-i\left(z+w\right)}=\frac{1}{2i}\left({e}^{iz}{e}^{iw}-{e}^{-iz}{e}^{-iw}\right).$

Again, both $f$ and $g$ are power series functions of the variable $w,$ and they agree on the sequence $\left\{1/k\right\}.$ Hence they agree everywhere, and this completes the proof of the first addition formula.

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:
${sin}^{2}\left(z\right)=\frac{1-cos\left(2z\right)}{2},$
and
${cos}^{2}\left(z\right)=\frac{1+cos\left(2z\right)}{2}.$

The trigonometric functions $sin$ and $cos$ are periodic with period $2\pi ;$ i.e., $sin\left(z+2\pi \right)=sin\left(z\right)$ and $cos\left(z+2\pi \right)=cos\left(z\right)$ for all complex numbers $z.$

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

$0={sin}^{2}\left(\pi \right)=\frac{1-cos\left(2\pi \right)}{2}$

which shows that $cos\left(2\pi \right)=1.$ Since ${cos}^{2}+{sin}^{2}=1,$ it then follows that $sin\left(2\pi \right)=0.$

The periodicity of the cosine function is proved similarly.

1. Prove that the hyperbolic functions $sinh$ and $cosh$ are periodic. What is the period?
2. Prove that the hyperbolic cosine $cosh\left(x\right)$ is never 0 for $x$ a real number, that the hyperbolic tangent $tanh\left(x\right)=sinh\left(x\right)/cosh\left(x\right)$ is bounded and increasing from $R$ onto $\left(-1,1\right),$ and that the inverse hyperbolic tangent has derivative given by ${{tanh}^{-1}}^{\text{'}}\left(y\right)=1/\left(1-{y}^{2}\right).$
3. Verify that for all $y\in \left(-1,1\right)$
${tanh}^{-1}\left(y\right)=ln\left(\sqrt{\frac{1+y}{1-y}}\right).$

Let $z$ be a nonzero complex number. Prove that there exists a unique real number $0\le \theta <2\pi$ such that $z=r{e}^{i\theta },$ where $r=|z|.$

HINT: If $z=a+bi,$ then $z=r\left(\frac{a}{r}+\frac{b}{r}i.$ Observe that $-1\le \frac{a}{r}\le 1,$ $-1\le \frac{b}{r}\le 1,$ and ${\left(\frac{a}{r}\right)}^{2}+{\left(\frac{b}{r}\right)}^{2}=1.$ Show that there exists a unique $0\le \theta <2\pi$ such that $\frac{a}{r}=cos\theta$ and $\frac{b}{r}=sin\theta .$

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
Got questions? Join the online conversation and get instant answers!
The content structure of the provided text is primarily expository, focusing on explaining the addition formulas for trigonometric and hyperbolic functions. It includes a detailed derivation of the sine addition formula and leaves the derivation of the other formulas as exercises for the reader. The text also includes several exercises and questions for the reader to solve.
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Given a text passage, determine:
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

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# 4.5 The trigonometric and hyperbolic functions

 Page 1 / 1
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 andhyperbolic functions, give the following “addition formulas:”

The following identities hold for all complex numbers $z$ and $w.$

$sin\left(z+w\right)=sin\left(z\right)cos\left(w\right)+cos\left(z\right)sin\left(w\right).$
$cos\left(z+w\right)=cos\left(z\right)cos\left(w\right)-sin\left(z\right)sin\left(w\right).$
$sinh\left(z+w\right)=sinh\left(z\right)cosh\left(w\right)+cosh\left(z\right)sinh\left(w\right).$
$cosh\left(z+w\right)=cosh\left(z\right)cosh\left(w\right)+sinh\left(z\right)sinh\left(w\right).$

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

First, for any two real numbers $x$ and $y,$ we have

$\begin{array}{ccc}\hfill cos\left(x+y\right)+isin\left(x+y\right)& =& {e}^{i\left(x+y\right)}\hfill \\ & =& {e}^{ix}{e}^{iy}\hfill \\ & =& \left(cosx+isinx\right)×\left(cosy+isiny\right)\hfill \\ & =& cosxcosy-sinxsiny+i\left(cosxsiny+sinxcosy\right),\hfill \end{array}$

which, equating real and imaginary parts, gives that

$cos\left(x+y\right)=cosxcosy-sinxsiny$

and

$sin\left(x+y\right)=sinxcosy+cosxsiny.$

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 holds for all complex numbers $z$ and $w.$ Thus, fix a real number $y.$ Let $f\left(z\right)=sinzcosy+coszsiny,$ and let

$g\left(z\right)=sin\left(z+y\right)=\frac{1}{2i}\left({e}^{i\left(z+y\right)}-{e}^{-i\left(z+y\right)}=\frac{1}{2i}\left({e}^{iz}{e}^{iy}-{e}^{-iz}{e}^{-iy}\right).$

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

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

$g\left(w\right)=sin\left(z+w\right)=\frac{1}{2i}\left({e}^{i\left(z+w\right)}-{e}^{-i\left(z+w\right)}=\frac{1}{2i}\left({e}^{iz}{e}^{iw}-{e}^{-iz}{e}^{-iw}\right).$

Again, both $f$ and $g$ are power series functions of the variable $w,$ and they agree on the sequence $\left\{1/k\right\}.$ Hence they agree everywhere, and this completes the proof of the first addition formula.

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:
${sin}^{2}\left(z\right)=\frac{1-cos\left(2z\right)}{2},$
and
${cos}^{2}\left(z\right)=\frac{1+cos\left(2z\right)}{2}.$

The trigonometric functions $sin$ and $cos$ are periodic with period $2\pi ;$ i.e., $sin\left(z+2\pi \right)=sin\left(z\right)$ and $cos\left(z+2\pi \right)=cos\left(z\right)$ for all complex numbers $z.$

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

$0={sin}^{2}\left(\pi \right)=\frac{1-cos\left(2\pi \right)}{2}$

which shows that $cos\left(2\pi \right)=1.$ Since ${cos}^{2}+{sin}^{2}=1,$ it then follows that $sin\left(2\pi \right)=0.$

The periodicity of the cosine function is proved similarly.

1. Prove that the hyperbolic functions $sinh$ and $cosh$ are periodic. What is the period?
2. Prove that the hyperbolic cosine $cosh\left(x\right)$ is never 0 for $x$ a real number, that the hyperbolic tangent $tanh\left(x\right)=sinh\left(x\right)/cosh\left(x\right)$ is bounded and increasing from $R$ onto $\left(-1,1\right),$ and that the inverse hyperbolic tangent has derivative given by ${{tanh}^{-1}}^{\text{'}}\left(y\right)=1/\left(1-{y}^{2}\right).$
3. Verify that for all $y\in \left(-1,1\right)$
${tanh}^{-1}\left(y\right)=ln\left(\sqrt{\frac{1+y}{1-y}}\right).$

Let $z$ be a nonzero complex number. Prove that there exists a unique real number $0\le \theta <2\pi$ such that $z=r{e}^{i\theta },$ where $r=|z|.$

HINT: If $z=a+bi,$ then $z=r\left(\frac{a}{r}+\frac{b}{r}i.$ Observe that $-1\le \frac{a}{r}\le 1,$ $-1\le \frac{b}{r}\le 1,$ and ${\left(\frac{a}{r}\right)}^{2}+{\left(\frac{b}{r}\right)}^{2}=1.$ Show that there exists a unique $0\le \theta <2\pi$ such that $\frac{a}{r}=cos\theta$ and $\frac{b}{r}=sin\theta .$

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
Got questions? Join the online conversation and get instant answers!
user
The content structure of the provided text is primarily expository, focusing on explaining the addition formulas for trigonometric and hyperbolic functions. It includes a detailed derivation of the sine addition formula and leaves the derivation of the other formulas as exercises for the reader. The text also includes several exercises and questions for the reader to solve.

TYPE: MULTIPLE_QUESTIONS