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

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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 S_m

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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 sin_mθ, cos_mθ, and tan_mθ for selected θ

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Theorem 2.2 For all , we have the following:

1.

2.

3.

Figure 3. Polar versus Cartesian coordinate

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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

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Figure 5.The graph of g-trigonometric functions when

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

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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 S_m in the first quadrant

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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

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.
	In article
[2]	Edmunds D. and Lang J., “Generalizing trigonometric functions from different points of view,” https://www.math.osu.edu/mri/preprints/2009, Tech. Rep., 2009.
	In article
[3]	Lang J. and Edmunds D., “Eigenvalues, Embeddings and Generalised Trigonometric Functions”, Springer, 2011.
	In article		CrossRef
[4]	Shelupsky D., “A generalization of trigonometric functions.”, The American Mathematical Monthly, Vol.66 no. 10, pp 879-884,1959.
	In article		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.
	In article