1)
Definitions
1.1 Right
triangle and unitcircle definitions
1.2 Definitions
via series
1.3 Definitions
via complex exponentials
1.4 Definitions
via differential equations
2)
A
difference between the real and complex
trigonometric functions
2.1 Relationship
to exponential function
2.2 The
complex sine and cosine functions are
not bounded
3)
Identities
3.1 Periodic
identities
3.2 Even
and odd identities
3.3 Pythagorean
identity
3.4 The
sum and difference formulas
3.5 The
doubleangle formulas
3.6 More
identities
4)
Calculus
1) Definitions
1.1
Right triangle and unitcircle definitions
The trigonometric functions
are most simply defined using the right
triangle. But the right triangle definitions
only define the trigonometric functions
for angles between 0 and
radians.
The six trigonometric
functions can also be defined in terms
of the unit circle (the circle of radius
one centered at the origin). The unit
circle definition permits the definition
of the trigonometric functions for all
positive and negative arguments.
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1.2 Definitions via series
The complex trigonometric
functions can be represented by the
power series:
Sin(z)
=
Cos(z)
=
Other complex trigonometric
functions are:
tan(z)
=
cot(z)
=
sec(z)
=
csc(z)
=
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1.3 Definitions via complex
exponentials
The complex trigonometric
functions can be defined algebraically
in terms of complex exponentials as:
Sin(z)
=
Cos(z)
=
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1.4 Definitions via differential
equations
Both the sine and cosine
functions satisfy the differential equation
y"
=  y
(1)
The
sine function is the unique solution
satisfying the initial conditions
y(0)
=
0 and y'(0)
=
1
Let
y = Sin(z)
Then
y'
= Cos(z);
y"
=  Sin(z)
Sin(z)
= Sin(z)
(1)
Initial conditions:
Sin(0)
= 0,
Sin'(0) = Cos(0)
= 1

The
cosine function is the unique
solution satisfying the initial
conditions y(0)
=
1 and y'(0)
=
0
Let
y = Cos(z)
Then
y'
=  Sin(z);
y"
=  Cos(z)
Cos(z)
= Cos(z)
(1)
Initial conditions:
Cos(0)
= 1,
Cos'(0) = Sin(0)
= 0

The
tangent function is the unique
solution of the nonlinear differential
equation
y'
= 1 + y^{2}
satisfying
the initial conditions y(0)
=
0
Let
y = tan(z)
Then
y'
= sec(z)^{2};
y^{2}
= tan(z)^{2}
sec(z)^{2}
= 1 + tan(z)^{2}
(1)
It
is an Pythagorean identity.
Initial conditions:
tan(0)
= 0

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2) A difference between the real and
complex trigonometric functions
There is a big difference
between the real and complex trigonometric
functions:
1) The real trigonometric
functions are not related to the exponential
function. But complex trigonometric
functions do have Relationship
to exponential function
2) The real sine and cosine
functions are bounded:
Sin(x)1,
Cos(x)1
The
complex sine and cosine functions are
not bounded if they
are defined over the set of all complex
numbers.
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2.1 Relationship to exponential
function
The main distinction between
real and complex trigonometric functions
is relationship to exponential function.
The complex trigonometric
functions can be defined algebraically
in terms of complex exponentials as:
Sin(z)
=
Cos(z)
=
It can be shown from the
series definitions that the sine and
cosine functions are the imaginary and
real parts, respectively, of the complex
exponential function when its argument
is purely imaginary:
This relationship was first noted by
Euler and the identity is called Euler's
formula.
The relationship between
the complex exponential and the trigonometric
functions can be expressed as:
Sin(z)
= =
iSinh(iz)
Cos(z)
= =
Cosh(iz)
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2.2 The complex sine and cosine
functions are not bounded
There is another distinction
between real and complex trigonometric
functions. In a case of complex variables
Sin(z)1
and
Cos(z)1
are not true.
For
example
Sin(i)
= 1.17520i,
Cos(i)
= 1.54308
=
1.3811
=
2.3811
But the Pythagorean
identity is true.
+=
1
+=
2.3811
 1.3811
= 1

Example:
Sin(z) = 2 Is it a paradox?
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3) Identities
3.1
Periodic identities
All complex trigonometric
functions are periodic functions with
the same periods as trigonometric function
for real variables.
The sine, cosine, secant, and cosecant
functions have period :
Sin(z
+)
= Sin(z)
Cos(z
+)
= Cos(z)
sec(z
+)
= sec(z)
csc(z
+)
= csc(z)
The tangent and cotangent
functions have period :
tan(z
+)
= tan(z)
cot(z
+)
= cot(z)
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3.2 Even and odd functions
Cos(z)
is an even function, Sin(z)
is an odd function as trigonometric
functions for real variables.
Sin(z)
=  Sin(z)
Cos(z)
= Cos(z)
sec(z)
= sec(z)
csc(z)
=  csc(z)
tan(z)
=  tan(z)
cot(z)
=  cot(z)
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3.3 Pythagorean identities
These identities are based
on the Pythagorean theorem.
+=
1
The second equation is obtained from
the first by dividing both sides by
.
tan(z)^{2}
+ 1
= sec(z)^{2}
The third equation is obtained from
the first by dividing both sides by
.
1
+ cot(z)^{2
}= csc(z)^{2}
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3.4 The sum and difference formulas
Other key relationships
are the sum and difference formulas,
which give the sine and cosine of the
sum and difference of two angles in
terms of sines and cosines of the angles
themselves.
Sin(z1+z2)
= Sin(z1)Cos(z2)
+ Cos(z1)Sin(z2)
Sin(z1z2)
= Sin(z1)Cos(z2)
 Cos(z1)Sin(z2)
Cos(z1+z2)
= Cos(z1)Cos(z2)
 Sin(z1)Sin(z2)
Cos(z1z2)
= Cos(z1)Cos(z2)
+ Sin(z1)Sin(z2)
When the two angles are
equal, the sum formulas reduce to simpler
equations known as the doubleangle
formulas.
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3.5 The doubleangle formulas
Sin(2z)
= 2Sin(z)Cos(z)
Cos(2z)
= 
3.6 More identities
Sin(z
+)
=  Sin(z)
Cos(z
+)
=  Cos(z)
Cos(z)
= Sin(z
+ )
Sin(
+ z)
= Sin(
 z)
The complex modulus satisfies
modulus identity:
Sin(x
+ iy)
= Sin(x)
+ Sin(iy)
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4) Calculus
If the sine and cosine
functions are defined by their Taylor
series, then the derivatives can be
found by differentiating the power series
termbyterm.
Sin(z)
= Cos(z)
Cos(z)
=  Sin(z)
The rest of the trigonometric
functions can be differentiated using
the above identities and the rules of
differentiation.
tan(z)
= sec(z)^{2}
cot(z)
=  csc(z)^{2}
sec(z)
= sec(z)tan(z)
csc(z)
=  csc(z)cot(z)
by
Tetyana Butler