Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

On Generalized Trigonometric Functions

Hisham Mahdi, Mohammed Elatrash, Samar ELmadhoun*

Department of Mathematics, Islamic University of Gaza, PO Box 108, Gaza, Palestine

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.

Keywords: trigonometric functions, generalized trigonometric functions, trigonometric identities

Received October 04, 2014; Revised November 27, 2014; Accepted December 05, 2014

Citation: Hisham Mahdi, Mohammed Elatrash, Samar ELmadhoun, On Generalized Trigonometric Functions. Journal of Mathematical Sciences and Applications, 2 (3). 33-38. January 2014. doi:10.12691/jmsa-2-3-2

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

Figure 1. Graph of for

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:

Figure 2. θ in standard position on Sm

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:

Table 1. Values of sinmθ, cosmθ, and tanmθ for selected θ

Theorem 2.2 For all , we have the following:

1.

2.

3.

Figure 3. Polar versus Cartesian coordinate

The polar equation of the unit semi-square is

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.

Figure 4.The graph of g-trigonometric functions when m=1

Figure 5.The graph of g-trigonometric functions when

Figure 6.The graph of g-trigonometric functions when m=4

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

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:

Figure 7. The graph of Sm in the first quadrant

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

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

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.
[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.
[3]   | Lang J. and Edmunds D., “Eigenvalues, Embeddings and Generalised Trigonometric Functions”, Springer, 2011.
[4]   | Shelupsky D., “A generalization of trigonometric functions.”, The American Mathematical Monthly, Vol.66 no. 10, pp 879-884,1959.
[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.