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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 ( z + w ) = sin ( z ) cos ( w ) + cos ( z ) sin ( w ) .
cos ( z + w ) = cos ( z ) cos ( w ) - sin ( z ) sin ( w ) .
sinh ( z + w ) = sinh ( z ) cosh ( w ) + cosh ( z ) sinh ( w ) .
cosh ( z + w ) = cosh ( z ) cosh ( w ) + sinh ( z ) sinh ( w ) .

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

First, for any two real numbers x and 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 ) ,

which, equating real and imaginary parts, gives that

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

and

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

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 ( z ) = sin z cos y + cos z sin y , 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 ) .

Then both f and g are power series functions of the variable z . Furthermore, by the previous calculation, f ( 1 / k ) = g ( 1 / k ) for all positive integers k . Hence, by the Identity Theorem, f ( z ) = g ( z ) 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 ( w ) = sin z cos w + cos z sin w , 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 ) .

Again, both f and g are power series functions of the variable w , and they agree on the sequence { 1 / k } . 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 ( z ) = 1 - cos ( 2 z ) 2 ,
    and
    cos 2 ( z ) = 1 + cos ( 2 z ) 2 .

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

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

0 = sin 2 ( π ) = 1 - cos ( 2 π ) 2

which shows that cos ( 2 π ) = 1 . Since cos 2 + sin 2 = 1 , it then follows that sin ( 2 π ) = 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 ( x ) is never 0 for x a real number, that the hyperbolic tangent tanh ( x ) = sinh ( x ) / cosh ( x ) is bounded and increasing from R onto ( - 1 , 1 ) , and that the inverse hyperbolic tangent has derivative given by tanh - 1 ' ( y ) = 1 / ( 1 - y 2 ) .
  3. Verify that for all y ( - 1 , 1 )
    tanh - 1 ( y ) = ln ( 1 + y 1 - y ) .

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

HINT: If z = a + b i , then z = r ( a r + b r i . Observe that - 1 a r 1 , - 1 b r 1 , and ( a r ) 2 + ( b r ) 2 = 1 . Show that there exists a unique 0 θ < 2 π such that a r = cos θ and b r = sin θ .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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