text: # 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!
explanation: The document is an excerpt from a mathematics text, specifically dealing with trigonometric and hyperbolic functions. It includes theorems and practice exercises related to the differentiation of these functions and their addition formulas. The text also provides a detailed derivation of the sine addition formula for complex numbers and leaves the derivation of other formulas as exercises for the reader. Additionally, it includes several exercises that require the reader to derive other formulas and prove properties of trigonometric and hyperbolic functions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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distilabel_metadata.role: system
distilabel_metadata.content: # 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!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an excerpt from a mathematics text, specifically dealing with trigonometric and hyperbolic functions. It includes theorems and practice exercises related to the differentiation of these functions and their addition formulas. The text also provides a detailed derivation of the sine addition formula for complex numbers and leaves the derivation of other formulas as exercises for the reader. Additionally, it includes several exercises that require the reader to derive other formulas and prove properties of trigonometric and hyperbolic functions.
  
  Label: multiple-question-and-answer