Abstract

In a previous paper (Gresham et al. 2019), the properties, theorems, and identities of the sine and cosine functions were developed using only analytical methods and without geometric constructions. We follow those results and use them to develop generalizations of the key theorems of trigonometry, again using purely analytical methods. We conclude with a connection to the Law of Conservation of Energy in physics.

classical analysis, trigonometry
In Gresham et al. (2019) a rigorous approach to the basic trig functions was developed where sin(x) was defined to be the solution of the 2nd order initial value problem (IVP)
formula
and cos(x) was defined to be the derivative of sin(x). Here we extend this technique to define generalized trigonometric functions.
Definition A generalized sine function, or sinusoid, is defined to be a nonconstant differentiable function which is a solution of the linear homogeneous differential equation
formula
for some k≠ 0.

The condition that k ≠ 0 ensures that f is not a linear function. The Existence and Uniqueness Theorem for Linear Homogeneous IVPs with constant coefficients (Nagle et al. 2008) tells us that such a solution exists, is unique, and has domain (−∞, ∞). In particular, it must be the case that for each real number t we must have f (t) ≠ 0 or f' (t) ≠ 0. If there is a value t where f (t) and f′ (t) are both 0, then the solution will be the identically constant 0 function. Therefore the function f is nonconstant if and only if a and b are not both 0.

The following theorems contains elementary consequences of the definition.

Theorem If f is a sinusoid and c ≠ 0 then f (x + c), f (c x), and c f (x) are sinusoids.

Proof: First note that since f is nonconstant, these three transformations of f are nonconstant. It remains to show that they satisfy the differential equation for a sinusoid. Since f is a sinusoid, there is a nonzero constant k such that
formula
for all (x) in (−∞, ∞). It suffices to show that
formula
where g is one of the above transformations. We consider each case separately.

Case I. Let g(x) = f (x+c). Then g′(x) = f′ (x+c) and g″(x) = f″ (x+c).

It follows that
formula
and the definition of a sinusoid is satisfied.

Case II. Let g(x) = f (cx).

Then g′(x) = cf′ (cx) and g″(x) = c2f″ (cx). It follows that
formula
Therefore
formula

Since c ≠ 0, we must have ck ≠ 0, and the definition of a sinusoid is satisfied.

Case III. Let g(x) = cf (x). Then g″(x) = cf″ (x) and
formula
It follows that
formula
and the definition of a sinusoid is satisfied. █

Theorem If f is a sinusoid then f′ is a sinusoid.

Proof: We first show that f′ satisfies the differential equation condition of the definition.

formula

Note that the initial conditions for f′ at 0 are f′(0) = b and f″(0) = −k2f (0) = −k2a. These are not both 0 since a and b are not both 0 and the number k ≠ 0. █

The following are consequences of this theorem.

Corollary If f is a sinusoid, then so is the nth derivative of f, denoted f[n].

Corollary If f is a sinusoid, then
formula
and
formula

The proofs of these are straightforward.

We denote a sinusoid f (x) with the notation
formula
where it is understood that k ≠ 0, and that sink,a,b (0) = a, and sin′k,a,b (0) = b, are not both 0.
We define its derivative to be
formula

We will see that

formula

In particular, the standard sine and cosine functions are the special cases

formula

Key theorems.–The two key theorems of plane trigonometry are the Pythagorean Identity and the Sine Sum Identity. With these two theorems one can develop all the other identities and values of special angles. In Gresham et al. (2019) it was shown that these can be rigorously proved without geometric constructions. Here we present a generalization of these two identities.

TheoremGeneralized Pythagorean Identity

If sink,a,b (x) is a sinusoid then for all x
formula
where c2 = (ka)2 + b2
Proof: We write sink,a,b(x) = f(x), = cosk,a,b(x) = f′(x) and consider
formula

The function f(x) has a second derivative for all real numbers x and so (k f (x))2 + (f′(x))2 satisfies the conditions of the Mean Value Theorem.

Its derivative is 0 for all x, and so (k f (x))2 + (f′ (x))2 must be a constant. Because k ≠ 0, and f (0) = a, and f′ (0) = b, with a and b are not both 0, this constant is
formula
█
We note that using sinusoids in parametric equations of the form
formula
will produce the graph of an ellipse.

We now turn our attention to the relationship between general sinusoids and the basic sine and cosine functions.

TheoremGeneral Representation of a Sinusoid

Sinusoids can be represented in terms of the standard sine and cosine functions as follows:
formula
Proof: Let f(x) = sink,a,b(x). We know that f is the solution of the IVP
formula

Let

Note that g(0) = a and since
formula
we have g′(0) = b. Then
formula

Therefore f and g satisfy the same IVP and by uniqueness must be equal. █

Using this General Representation theorem, we can produce a representation for cosk,a,b(x).

Corollary cosk,a,b(x) = sink,b,−k2a(x).

Proof:
formula
█
The Sine Sum formula, whose non-geometric proof is included in Gresham et al. (2019),
formula
serves as a key identity in the development of classical trigonometry.

To prepare for a generalization of this identity in terms of sink,a,b(x) and cosk,a,b(x), we next consider a representation of sink,a,b(x) as a transformation of the classical sine function.

LemmaTransformation Representation Let z be the complex number ⁠. Then
formula
where arg z is the argument of the complex number z.
Proof: With ⁠, we have
formula
It follows then that
formula
█

Using this result and the Sine Sum formula, we can write the Sinusoid Sum formula.

TheoremSinusoid Sum Formula

If f(x) = sink,a,b(x) is a sinusoid and ⁠, then
formula

Proof: In this proof we will view y as an arbitrary fixed real constant. We will show that the left and right sides of the equation above are sinusoids in x with the same initial conditions.

Observe that since f is a sinusoid, the left side of the equation, a horizontal transformation of f, is a sinusoid and must satisfy the condition
formula
To simplify our analysis of the right side, let g (x) represent the right side, again viewing y as a constant:
formula
Now we find the first and second derivatives of the right side, treating y as a constant.
formula
and
formula
From this we see that g satisfies the DE for a sinusoid. What are the initial values g(0) and g′(0)? We first do some preliminary calculations using the Transformation Representation Theorem previously proved. Note that
formula
We will need these two evaluations:
formula
and
formula
Now we are ready to calculate g (0) and g′(0):
formula
and
formula

Since f (x + y) is a sinusoid in x, f (0 + y) and f′ (0 + y) are not both 0. Then f (x + y) and g (x) are both sinusoids in x with the same initial conditions, and therefore by Uniqueness they are equal. █

Note Letting f (x) = sink,a,b(x), f′(x) = cosk,a,b(x), and ⁠, the Sinusoid Sum formula can be written as
formula

Corollary The sine and cosine sum formulas are special cases of the General Sinusoid Sum Formula.

Proof: If k = 1, a = 0, and b = 1, then f (x) = sin(x) and z = 1+0i. The General Sum Formula reduces to
formula
If k = 1, a = 1, and b = 0, then f (x) = cos(x) and z = 0 +1 i. The General Formula reduces to
formula
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We now observe that with additional conditions, namely that the function f be analytic, the converse of the General Pythagorean Identity also holds.

Theorem If f (x) is nonconstant, analytic on (−∞, ∞), and for all real x satisfies the Generalized Pythagorean Identity
formula
where k and c are constants, then f is a sinusoid.
Proof: Differentiation of both sides gives
formula
so that
formula
Since f is nonconstant, f′ (d) ≠ 0 for some number d. Because f is analytic, f′ is continuous and there exists an open interval S containing d such that f′ (x) is nonzero on S. On S it must be the case that
formula

The function f is a solution of the differential equation for a sinusoid. Since such solutions are unique, it must follow that f is the solution with domain (−∞, ∞). The function f is nonconstant and it must follow that f(0) and f′(0) are not both 0. Therefore the function f is a sinusoid. █

Connection with the Law of Conservation of Energy.–Simple Harmonic Motion (SHM) is used to describe a system in which an object of mass m experiences a restoring force which is directly proportional to its displacement from its equilibrium position and there are no other forces such as damping forces or friction involved. The common applications are pendulums with small displacements or an oscillating mass on a spring. We will consider a system with a spring with spring constant k connected to an object of mass m with no other forces involved. We know from Hooke’s Law that in such a system the restoring force is described by
formula
where x = x(t) is a one-dimensional vector which measures its displacement from the equilibrium position at time t.
In the SI system, the magnitude of this one-dimensional force vector is measured in Newtons (N), the spring constant k > 0 is measured in Newtons/meter(N/m), and the magnitude of displacement vector x(t) is measured in meters. The spring potential energy (SPE), measured in Joules (J), in the system is
formula
and if v is the magnitude of the one-dimensional velocity vector of the mass, the kinetic energy (KE) in the system is
formula
Our goal here is two-fold: first to describe the position x(t) of the mass at time t, and second, to describe the energy of the system. We begin with a solution for the differential equation given by Hooke’s Law.
formula
Since
formula
we can write the differential equation for SHM
formula
From what we have seen earlier, the solution of this system is
formula
Next, we consider the connection between the Generalized Pythagorean Identity for this sinusoid and the energy of the system. We know that at any time t
formula
From substituting the initial conditions at t = 0,
formula

We now write this last expression in terms of the SPE and KE of the system.

formula

From the Law of Conservation of Energy we know that the total energy of the harmonic oscillator is constant. Therefore, the Generalized Pythagorean Theorem produces a constant which is twice the energy of the system (in Joules) divided by the mass (in kg) of the system. The energy of a system divided by its mass is called the specific energy of the system.

We can now state the result of this discussion.

Theorem When the motion of a mass attached to a spring in a frictionless system is described by a sinusoid, then the constant c2 in the Generalized Pythagorean Identity for this sinusoid is twice the specific energy of the system.

Summary and Conclusions.–We have extended the theorems and identities of basic trigonometry using the definition of a generalized sine function, called a sinusoid, as the solution of a certain second order linear homogeneous differential equation. The sine function itself can be defined as a solution for particular initial values, and expressed in terms of a power series. The key theorems in this study are the Pythagorean Identity and the Sine Sum Identity, which we have stated in a more general form here. The Pythagorean Identity, in its generalized form, characterizes sinusoid functions as well as providing a connection to physics.

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