On Generalized Trigonometric Functions
Hisham Mahdi1, Mohammed Elatrash1, Samar ELmadhoun1,
1Department of Mathematics, Islamic University of Gaza, PO Box 108, Gaza, Palestine
2. Generalized Trigonometric Functions; Definitions and Graphs
3. Identities and Some Common Properties
Abstract
A new trigonometric functions called generalized trigonometric functions are perversely defined by a system of first order nonlinear ordinary differential equations with initial conditions. This system is related to the Hamilton system. In this paper, we define these functions using the equation , for m>0 We study the graphs, the trigonometric identities and some of common properties of these functions. We find the first derivatives which have different forms when is even and when is odd.
At a glance: Figures
Keywords: trigonometric functions, generalized trigonometric functions, trigonometric identities
Journal of Mathematical Sciences and Applications, 2014 2 (3),
pp 33-38.
DOI: 10.12691/jmsa-2-3-2
Received October 04, 2014; Revised November 27, 2014; Accepted December 05, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.Cite this article:
- Mahdi, Hisham, Mohammed Elatrash, and Samar ELmadhoun. "On Generalized Trigonometric Functions." Journal of Mathematical Sciences and Applications 2.3 (2014): 33-38.
- Mahdi, H. , Elatrash, M. , & ELmadhoun, S. (2014). On Generalized Trigonometric Functions. Journal of Mathematical Sciences and Applications, 2(3), 33-38.
- Mahdi, Hisham, Mohammed Elatrash, and Samar ELmadhoun. "On Generalized Trigonometric Functions." Journal of Mathematical Sciences and Applications 2, no. 3 (2014): 33-38.
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1. Introduction
Ordinary trigonometry studies triangles in the Euclidean plane . There are some ways to defining the ordinary trigonometric functions on real numbers such as right-angled triangle definition, unit-circle definition, series definition, definitions via differential equations, and definition using functional equations. Trigonometric functions are one of the important group of the elementary functions. Using them, we can solve geometric problems, complex analytic problems and problems involving Fourier series. Also they are important because they are periodic. All the six trigonometric functions can defined through the sine and cosine functions.
In many papers, (see [1, 2, 3, 4]), a new trigonometric functions are defined using a system of first order nonlinear ordinary differential equations with initial conditions. This system is related to the Hamilton system. The new functions are called generalized trigonometric functions and denoted by
for
It was proved that if
and
then
In this paper, we define these functions directly using the equation
for
We study the graphs and the trigonometric identities of these functions. Then we study the first derivative for special cases when
is natural number. Since trigonometric functions are used in Fourier series, Fourier transform, and signal processing, we look to improve the efficiency of signal processing and reduce the noice effects by using the generalized trigonometric functions. Moreover, the generalized trigonometric functions can be used to obtain analytic solutions to the equation of a nonlinear spring-max system.
Now, consider the equation The graph of this equation in the Cartesian plan is symmetric about the axes. For a special case, if
, the graph of
is a unit square centered at
with vertices at
For
the graph is the unit circle. For any
let
be the graph of the equation
We call
a unit semi-square. For an angle
placed in the center of
with initial ray on the positive
and terminal ray intersects the graph of
n a point
, we say that
is placed in the standard position of
. Suppose that the angle
is placed in the standard position of
and
s the point of intersection of the terminal ray and
. Then along the terminal ray, we have the following:
1. as
and
2. as
In Figure 1, we graph the equation for several values of
showing the point


Definition 1.1 In the Cartesian plane, if is an angle placed in the center of the plane with initial ray at the positive
then we say that:
1. if the terminal ray lies between the positive
and the positive
if
or
2. if the terminal ray lies between the negative
and the positive
if
or
3. if the terminal ray lies between the negative
and the negative
if
or
4. if the terminal ray lies between the positive
and the negative
if
or
2. Generalized Trigonometric Functions; Definitions and Graphs
Definition 2.1 For a given and for a unit semi-square
let
be an angle placed in the standard position. Suppose that the terminal ray intersects
in a point
(as seenin Figure 2 ). We define the six generalized trigonometric functions of
as follows:
1. g-sine of
2. g-cosine of
3. g-tangent of
4. g-cosecant of
5. g-secant of
6. g-cotangent of
The following table gives some values of g-trigonometric functions for some special angles:
Theorem 2.2 For all , we have the following:
1.
2.
3.
The polar equation of the unit semi-square is
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Using this equation,we get relations between usual trigonometric and g- functions in the following theorem:
Theorem 2.3 For any we have the following:
1.
2.
3.
4.
5.
6.
Corollary 2.4 Let be a g-trigonometric functions.For any
and for any
we have that
That is, g-trigonometric functions are periodic functions.
Theorem 2.5 Let . Then:
1.
2.
3.
4.
Using the relations between g-trigonometric functions and usual trigonometric functions and using the Graph-4.4.2 grapher program, we give graphs of g-trigonometric functions for and
We neglect the graphs of
and
since they are exactly the graphs of
and
respectively.

3. Identities and Some Common Properties
Evidently, and for , the g-trigonometric functions have the following direct common properties:
1.
2. All g-trigonometric functions are periodic. Moreover, the functions and
have period
, while the functions
and
have period
3. The g-trigonometric functions and
are even functions, while the other g-trigonometric functions are odd functions.
Theorem 3.1 For any and
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Proof. Consider the graph of
in the first quadrant, as shown Figure 7. Draw the two vectors
and
as terminal rays of the two angles
and
respectively. So, we have the following:
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Hence
![]() |
Corollary 3.2 For any
1.
2.
The proof of the following theorem can be done directly or similar to the proof ofTheorem 3.1. Either way, it is easy and we will omit it.
Theorem 3.3 For any
1.
2.
3.
4.
5.
6.
7.
Theorem 3.4 For any we have that
4. Derivatives of g-Trigonometric Functions
In this section and unless otherwise statement, we consider as a natural number. Let
be an even natural number and
be a unit semi-square. Then the polar equation of
has the form
So, the first derivative of
with respect to
is
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Theorem 4.1 If is even, then
1.
2.
3.
4.
5.
6.
Mathematically, if is even, then
But if
is odd, the value of
depends on the sign of
. So, there is a quite difference between the forms of derivatives of the g-trigonometric functions when
is even and when m is odd. Moreover, we will see that in the case when
is odd, the derivatives of the g-trigonometric functions have different forms when
and when
If
is odd, the polar equation of
has the following piecewise definition function with four cases depending on
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In order to simplify the derivatives of the generalized trigonometric functions when is odd, define the following four functions:
![]() |
So, we have that for if
then
and if
then
Theorem 4.2 Let be an odd natural number, and let
be the polar equation of
Then for
![]() |
Proof. Derive directly, we get that
![]() |
As is odd,
is even and we have that
![]() |
Since
![]() |
Theorem 4.3 If then
does not exist for all
Proof. For the right hand and the left hand derivatives of
with respect to
at
are
![]() |
Hence, does not exist for all
,
Similarly, for all
Theorem 4.4 If is odd, and
then
for all
.
Proof. If then
both
and
This implies that
![]() |
Remark 4.5 For all and for
we have
and
so we get that
![]() |
Moreover, if and at
we have that,
and
have values 1 or -1, and
Hence
![]() |
Using this remark, we have the following summary theorem.
Theorem 4.6 Let be an odd natural number, and let
be the polar equation of the unit semi-square
Then
a. For and for
![]() |
b. For and for all
![]() |
Theorem 4.7
a. For and for
1.
2.
3.
4.
b. For and for all
1.
2.
3.
4.
Proof. a) (1) For nd for
we have
![]() |
b)(1) For we have two cases:
Case 1:
In this case, and
So,
![]() |
Since is odd,
is even, so
![]() |
Case 2: In this case,
So,
![]() |
Now if then
In this case,
![]() |
And if In this case,
![]() |
In both cases, we have that
![]() |
Remark 4.8 Since and
we have that
![]() |
References
[1] | Burgoyre F.D., “Generalization trigonometric functions.”, Mathematics of Computation, vol.18, pp 3-14-316,1964. | ||
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[2] | Edmunds D. and Lang J., “Generalizing trigonometric functions from different points of view,” http://www.math.osu.edu/mri/preprints/2009, Tech. Rep., 2009. | ||
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[3] | Lang J. and Edmunds D., “Eigenvalues, Embeddings and Generalised Trigonometric Functions”, Springer, 2011. | ||
![]() | CrossRef | ||
[4] | Shelupsky D., “A generalization of trigonometric functions.”, The American Mathematical Monthly, Vol.66 no. 10, pp 879-884,1959. | ||
![]() | CrossRef | ||
[5] | Wei D.,Elgindi Y. L..,and Elgindi M. B., “Some Generalized Trigonometric Sine Functions and Their Applications”, Applied Mathematical Sciences, Vol. 6, no. 122, 6053-6068, 2012. | ||
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