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.
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.
Case I. Let g(x) = f (x+c). Then g′(x) = f′ (x+c) and g″(x) = f″ (x+c).
Case II. Let g(x) = f (cx).
Since c ≠ 0, we must have ck ≠ 0, 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.
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].
The proofs of these are straightforward.
We will see that
In particular, the standard sine and cosine functions are the special cases
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
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.
We now turn our attention to the relationship between general sinusoids and the basic sine and cosine functions.
TheoremGeneral Representation of a Sinusoid
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).
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.
Using this result and the Sine Sum formula, we can write the Sinusoid Sum formula.
TheoremSinusoid Sum 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.
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. █
Corollary The sine and cosine sum formulas are special cases of the General Sinusoid Sum Formula.
We now observe that with additional conditions, namely that the function f be analytic, the converse of the General Pythagorean Identity also holds.
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. █
We now write this last expression in terms of the SPE and KE of the system.
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.