trigonometric and hyperbolic functions
TRANSCRIPT
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Trigonometric and Hyperbolic Functions
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction
1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).
(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).
2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).
(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).
(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).
(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).
(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Introduction1. For real numbers θ we have eiθ = cos(θ)+ isin(θ).2. Replacing θ with −θ we obtain e−iθ = cos(θ)− isin(θ).
(Remember that the cosine is even and the sine is odd.)
3. Adding the two and dividing by 2 gives cos(θ) =eiθ + e−iθ
2.
4. Subtracting the two and dividing by 2i gives sin(θ) =eiθ − e−iθ
2i.
5. The right sides above make sense for all complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition.
For any complex number z we define
cos(z) =eiz + e−iz
2and
sin(z) =eiz− e−iz
2i.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cos(z) =eiz + e−iz
2
and
sin(z) =eiz− e−iz
2i.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cos(z) =eiz + e−iz
2and
sin(z) =eiz− e−iz
2i.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
This Time, All Common Identities Carry Over
1. sin(z1 + z2) = sin(z1)cos(z2)+ cos(z1)sin(z2)2. cos(z1 + z2) = cos(z1)cos(z2)− sin(z1)sin(z2)3. sin2(z)+ cos2(z) = 14. sin(z+2π) = sin(z)5. cos(z+2π) = cos(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
This Time, All Common Identities Carry Over1. sin(z1 + z2) = sin(z1)cos(z2)+ cos(z1)sin(z2)
2. cos(z1 + z2) = cos(z1)cos(z2)− sin(z1)sin(z2)3. sin2(z)+ cos2(z) = 14. sin(z+2π) = sin(z)5. cos(z+2π) = cos(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
This Time, All Common Identities Carry Over1. sin(z1 + z2) = sin(z1)cos(z2)+ cos(z1)sin(z2)2. cos(z1 + z2) = cos(z1)cos(z2)− sin(z1)sin(z2)
3. sin2(z)+ cos2(z) = 14. sin(z+2π) = sin(z)5. cos(z+2π) = cos(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
This Time, All Common Identities Carry Over1. sin(z1 + z2) = sin(z1)cos(z2)+ cos(z1)sin(z2)2. cos(z1 + z2) = cos(z1)cos(z2)− sin(z1)sin(z2)3. sin2(z)+ cos2(z) = 1
4. sin(z+2π) = sin(z)5. cos(z+2π) = cos(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
This Time, All Common Identities Carry Over1. sin(z1 + z2) = sin(z1)cos(z2)+ cos(z1)sin(z2)2. cos(z1 + z2) = cos(z1)cos(z2)− sin(z1)sin(z2)3. sin2(z)+ cos2(z) = 14. sin(z+2π) = sin(z)
5. cos(z+2π) = cos(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
This Time, All Common Identities Carry Over1. sin(z1 + z2) = sin(z1)cos(z2)+ cos(z1)sin(z2)2. cos(z1 + z2) = cos(z1)cos(z2)− sin(z1)sin(z2)3. sin2(z)+ cos2(z) = 14. sin(z+2π) = sin(z)5. cos(z+2π) = cos(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)
=ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i
= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Some Proofs Actually Are Simpler Now
sin(z1)cos(z2)+ cos(z1)sin(z2)
=eiz1− e−iz1
2ieiz2 + e−iz2
2+
eiz1 + e−iz1
2eiz2− e−iz2
2i
=14i
(ei(z1+z2) + ei(z1−z2)− ei(−z1+z2)− e−i(z1+z2)
+ ei(z1+z2)− ei(z1−z2) + ei(−z1+z2)− e−i(z1+z2))
=14i
(2ei(z1+z2)−2e−i(z1+z2)
)=
ei(z1+z2)− e−i(z1+z2)
2i= sin(z1 + z2)
This is why many people like to work with the complex definition ofthe trigonometric functions.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Other Proofs Stay The Same
sin(z+2π) = sin(z)cos(2π)+ cos(z)sin(2π)= sin(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Other Proofs Stay The Same
sin(z+2π)
= sin(z)cos(2π)+ cos(z)sin(2π)= sin(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Other Proofs Stay The Same
sin(z+2π) = sin(z)cos(2π)+ cos(z)sin(2π)
= sin(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Other Proofs Stay The Same
sin(z+2π) = sin(z)cos(2π)+ cos(z)sin(2π)= sin(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Other Trigonometric Functions
1. tan(z) :=sin(z)cos(z)
2. cot(z) :=cos(z)sin(z)
3. sec(z) :=1
cos(z)
4. csc(z) :=1
sin(z)Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Other Trigonometric Functions1. tan(z) :=
sin(z)cos(z)
2. cot(z) :=cos(z)sin(z)
3. sec(z) :=1
cos(z)
4. csc(z) :=1
sin(z)Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Other Trigonometric Functions1. tan(z) :=
sin(z)cos(z)
2. cot(z) :=cos(z)sin(z)
3. sec(z) :=1
cos(z)
4. csc(z) :=1
sin(z)Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Other Trigonometric Functions1. tan(z) :=
sin(z)cos(z)
2. cot(z) :=cos(z)sin(z)
3. sec(z) :=1
cos(z)
4. csc(z) :=1
sin(z)Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Other Trigonometric Functions1. tan(z) :=
sin(z)cos(z)
2. cot(z) :=cos(z)sin(z)
3. sec(z) :=1
cos(z)
4. csc(z) :=1
sin(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Other Trigonometric Functions1. tan(z) :=
sin(z)cos(z)
2. cot(z) :=cos(z)sin(z)
3. sec(z) :=1
cos(z)
4. csc(z) :=1
sin(z)Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)
=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i
=ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i
=eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2
=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
The Derivatives
1.ddz
sin(z)=ddz
eiz−e−iz
2i=
ieiz+ie−iz
2i=
eiz+e−iz
2=cos(z)
2.ddz
cos(z) =−sin(z)
3.ddz
tan(z) =1
cos2(z)
4.ddz
cot(z) =− 1sin2(z)
5.ddz
sec(z) = sec(z) tan(z)
6.ddz
csc(z) =−csc(z)cot(z)
Note that secant and cosecant are not very common (around theworld).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition.
For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2
and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz)
=eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i
=e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i
= iiez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i
= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz)
= cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Definition. For any complex number z we define
cosh(z) =ez + e−z
2and
sinh(z) =ez− e−z
2.
Note.
sin(iz) =eiiz− e−iiz
2i=
e−z− ez
2i= ii
ez− e−z
2i= isinh(z)
cos(iz) = cosh(z)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine
|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)
=(
sin(x)cos(iy)+ sin(iy)cos(x))sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))
××(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)
+sinh2(y)sin2(x)− sinh2(y)sin2(x)= sin2(x)
(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)
= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Another Connection Between Sine and Hyperbolic Sine|sin(z)|2
= sin(x+ iy)sin(x+ iy)=
(sin(x)cos(iy)+ sin(iy)cos(x)
)sin(x+ iy)
=(
sin(x)cosh(y)+ isinh(y)cos(x))×
×(
sin(x)cosh(y)− isinh(y)cos(x))
= sin2(x)cosh2(y)+ sinh2(y)cos2(x)+sinh2(y)sin2(x)− sinh2(y)sin2(x)
= sin2(x)(cosh2(y)− sinh2(y)
)+ sinh2(y)
(cos2(x)+ sin2(x)
)= sin2(x)+ sinh2(y)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.
The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof.
The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2
=∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2
= sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0
and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ .
So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex sine function are thenumbers nπ where n is an integer.
Proof. The only solutions of
0 =∣∣sin(z)
∣∣2 =∣∣sin(x+ iy)
∣∣2 = sin2(x)+ sinh2(y)
are y = 0 and x = nπ . So z = nπ where n is an integer are the zeros ofthe sine function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions
1.ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)
4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)
5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 1
6. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)
7. cosh(z+2πi) = cosh(z)... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more.
And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Identities for Hyperbolic Functions1.
ddz
sinh(z) = cosh(z)
2.ddz
cosh(z) = sinh(z)
3. sinh(z1 + z2) = sinh(z1)cosh(z2)+ cosh(z1)sinh(z2)4. cosh(z1 + z2) = cosh(z1)cosh(z2)+ sinh(z1)sinh(z2)5. cosh2(z)− sinh2(z) = 16. sinh(z+2πi) = sinh(z)7. cosh(z+2πi) = cosh(z)
... and more. And they can all be verified straight from the definition.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.
The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof. sinh(z) =−isin(iz) is equal to zero where sin(iz) is equal tozero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof. sinh(z) =−isin(iz) is equal to zero where sin(iz) is equal tozero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof.
sinh(z) =−isin(iz) is equal to zero where sin(iz) is equal tozero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof. sinh(z) =−isin(iz)
is equal to zero where sin(iz) is equal tozero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof. sinh(z) =−isin(iz) is equal to zero where sin(iz) is equal tozero.
Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof. sinh(z) =−isin(iz) is equal to zero where sin(iz) is equal tozero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. The only zeros of the complex hyperbolic sine function arethe numbers nπi, where n is an integer.
Proof. sinh(z) =−isin(iz) is equal to zero where sin(iz) is equal tozero. Hence the zeros of sinh(z) are at z = nπi, where n is an integer.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
More Hyperbolic Functions
1. tanh(z) :=sinh(z)cosh(z)
2. coth(z) :=cosh(z)sinh(z)
3. sech(z) :=1
cosh(z)
4. csch(z) :=1
sinh(z)Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
More Hyperbolic Functions1. tanh(z) :=
sinh(z)cosh(z)
2. coth(z) :=cosh(z)sinh(z)
3. sech(z) :=1
cosh(z)
4. csch(z) :=1
sinh(z)Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
More Hyperbolic Functions1. tanh(z) :=
sinh(z)cosh(z)
2. coth(z) :=cosh(z)sinh(z)
3. sech(z) :=1
cosh(z)
4. csch(z) :=1
sinh(z)Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
More Hyperbolic Functions1. tanh(z) :=
sinh(z)cosh(z)
2. coth(z) :=cosh(z)sinh(z)
3. sech(z) :=1
cosh(z)
4. csch(z) :=1
sinh(z)Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
More Hyperbolic Functions1. tanh(z) :=
sinh(z)cosh(z)
2. coth(z) :=cosh(z)sinh(z)
3. sech(z) :=1
cosh(z)
4. csch(z) :=1
sinh(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
More Hyperbolic Functions1. tanh(z) :=
sinh(z)cosh(z)
2. coth(z) :=cosh(z)sinh(z)
3. sech(z) :=1
cosh(z)
4. csch(z) :=1
sinh(z)Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Remaining Derivatives
1.ddz
tanh(z) =1
cosh2(z)
2.ddz
coth(z) =− 1sinh2(z)
3.ddz
sech(z) =−sech(z) tanh(z)
4.ddz
csch(z) =−csch(z)coth(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Remaining Derivatives1.
ddz
tanh(z) =1
cosh2(z)
2.ddz
coth(z) =− 1sinh2(z)
3.ddz
sech(z) =−sech(z) tanh(z)
4.ddz
csch(z) =−csch(z)coth(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Remaining Derivatives1.
ddz
tanh(z) =1
cosh2(z)
2.ddz
coth(z) =− 1sinh2(z)
3.ddz
sech(z) =−sech(z) tanh(z)
4.ddz
csch(z) =−csch(z)coth(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Remaining Derivatives1.
ddz
tanh(z) =1
cosh2(z)
2.ddz
coth(z) =− 1sinh2(z)
3.ddz
sech(z) =−sech(z) tanh(z)
4.ddz
csch(z) =−csch(z)coth(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Remaining Derivatives1.
ddz
tanh(z) =1
cosh2(z)
2.ddz
coth(z) =− 1sinh2(z)
3.ddz
sech(z) =−sech(z) tanh(z)
4.ddz
csch(z) =−csch(z)coth(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Remaining Derivatives1.
ddz
tanh(z) =1
cosh2(z)
2.ddz
coth(z) =− 1sinh2(z)
3.ddz
sech(z) =−sech(z) tanh(z)
4.ddz
csch(z) =−csch(z)coth(z)
Note that hyperbolic secant and cosecant are not very common(around the world) either.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.
For all complex numbers z we have
arccos(z) =−i log(
z+ i(1− z2) 1
2
)where the right side is a multivalued function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. For all complex numbers z we have
arccos(z) =−i log(
z+ i(1− z2) 1
2
)
where the right side is a multivalued function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem. For all complex numbers z we have
arccos(z) =−i log(
z+ i(1− z2) 1
2
)where the right side is a multivalued function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)
cos(w) = zeiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0
(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Proof.
w = arccos(z)cos(w) = z
eiw + e−iw
2= z
eiw−2z+ e−iw = 0(eiw)2−2zeiw +1 = 0
eiw =2z±√
4z2−42
= z±√
z2−1
w =1i
log(
z±√
z2−1)
= −i log(
z± i√
1− z2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly We Can Derive ...
1. arcsin(z) =−i log(
iz+(1− z2) 1
2
)2. arctan(z) =
i2
log(
i+ zi− z
)3. arsinh(z) = log
(z+(1+ z2) 1
2
)4. arcosh(z) = log
(z+(z2−1
) 12
)5. artanh(z) =
12
log(
1+ z1− z
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly We Can Derive ...1. arcsin(z) =−i log
(iz+
(1− z2) 1
2
)
2. arctan(z) =i2
log(
i+ zi− z
)3. arsinh(z) = log
(z+(1+ z2) 1
2
)4. arcosh(z) = log
(z+(z2−1
) 12
)5. artanh(z) =
12
log(
1+ z1− z
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly We Can Derive ...1. arcsin(z) =−i log
(iz+
(1− z2) 1
2
)2. arctan(z) =
i2
log(
i+ zi− z
)
3. arsinh(z) = log(
z+(1+ z2) 1
2
)4. arcosh(z) = log
(z+(z2−1
) 12
)5. artanh(z) =
12
log(
1+ z1− z
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly We Can Derive ...1. arcsin(z) =−i log
(iz+
(1− z2) 1
2
)2. arctan(z) =
i2
log(
i+ zi− z
)3. arsinh(z) = log
(z+(1+ z2) 1
2
)
4. arcosh(z) = log(
z+(z2−1
) 12
)5. artanh(z) =
12
log(
1+ z1− z
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly We Can Derive ...1. arcsin(z) =−i log
(iz+
(1− z2) 1
2
)2. arctan(z) =
i2
log(
i+ zi− z
)3. arsinh(z) = log
(z+(1+ z2) 1
2
)4. arcosh(z) = log
(z+(z2−1
) 12
)
5. artanh(z) =12
log(
1+ z1− z
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly We Can Derive ...1. arcsin(z) =−i log
(iz+
(1− z2) 1
2
)2. arctan(z) =
i2
log(
i+ zi− z
)3. arsinh(z) = log
(z+(1+ z2) 1
2
)4. arcosh(z) = log
(z+(z2−1
) 12
)5. artanh(z) =
12
log(
1+ z1− z
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.
ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
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Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.
ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z)
=ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))
= −i1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Theorem.ddz
arcsin(z) =1
(1− z2)12
where the right side is
multivalued again.
Proof.ddz
arcsin(z) =ddz
(−i log
(iz+
(1− z2) 1
2
))= −i
1
iz+(1− z2)12
(i+
12
1
(1− z2)12(−2z)
)
=−i
iz+(1− z2)12
i(1− z2
) 12 − z
(1− z2)12
=
1
(1− z2)12
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly ...
1.ddz
arccos(z) =− 1
(1− z2)12
2.ddz
arctan(z) =1
1+ z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly ...1.
ddz
arccos(z) =− 1
(1− z2)12
2.ddz
arctan(z) =1
1+ z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions
logo1
Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions
Similarly ...1.
ddz
arccos(z) =− 1
(1− z2)12
2.ddz
arctan(z) =1
1+ z2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Trigonometric and Hyperbolic Functions