Abstract

The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous approach. Our purpose here is to revisit elementary trigonometry from an entirely analytic perspective. We will give a comprehensive treatment of the sine and cosine functions and will show how to derive the familiar theorems of trigonometry without reference to geometric definitions or constructions.

Supplemental material is available for this article online.

Our purpose in this paper is to show how the definitions and theorems of elementary trigonometry can be developed without references to geometric constructions. We will use methods from real analysis to provide an alternate view of the sine and cosine functions. Along the way we will see a relationship that leads to a non-geometric construction of pi. Finally, we will make connections with the familiar geometric approach. For this study, we will assume a familiarity with calculus, differential equations, and real analysis. Since simple harmonic motion (SHM) of an oscillator follows a sinusoidal pattern, we will use the differential equation for SHM as the basis for our development of the sine and cosine functions.

Definitions and basic properties.–We begin by considering the solution of the second-order homogeneous linear differential equation

formula

By the Existence and Uniqueness Theorem we know that a unique solution exists (Nagle et al. 2008). If this solution has a power series representation around the ordinary point x = 0, it must have the form

formula

Note that f (0) = c0 = 0 and f′ (0) = c1 = 1. We also have

formula

then

formula

Since this power series is 0 for all x, we get the general recursion relation

formula

so that

formula

because c0 = 0, we have for all even indices 2n

formula

Let us now examine the coefficients with odd indices 2n + 1.

formula

and in general,

formula

The power series about x = 0 must have the form

formula

Using the Ratio Test, it is easy to show that this series converges for all real x. The function represented by this power series is the unique solution of the differential equation

formula

We call this function the sine function, denoted sin x, or sin(x).

Definition Sine Function

formula

We define the cosine to be the derivative of the sine function.

Definition Cosine Function

formula

The following are elementary consequences of the definitions:

  1. sin(0) = 0

  2. cos(0) = 1

  3. The function sin(x) is odd because all exponents in its power series are odd.

  4. The function cos(x) is even because all exponents in its power series are even.

  5. The functions sin(x) and and cos(x) are both continuous since they are differentiable.

  6. The derivatives of sin(x) are cyclic with order four.

Key theorems.–This section presents the Pythagorean and Sine Sum identities which, along with the smallest positive critical value of sin x, enable the development of several important identities and analytic results in elementary trigonometry.

First, we prove the Pythagorean Identity.

Theorem Pythagorean Identity For all x,

formula

Proof: Consider the derivative of the left side.

formula

Since the derivative is 0, sin2 x + cos2x is a constant.

Because sin(0) = 0, and cos(0) = 1, this constant must be 1.

Next, we consider the identity for the sine of the sum of x and y. The proof in most elementary trigonometry texts involves a geometric construction with triangles or the unit circle. In our geometry-free approach, we will use only power series.

Theorem Sine Sum Identity For all x, y,

formula

Proof: Consider the series expansion

formula

Now examine the general nth term an of this series using the Binomial Theorem:

formula

This last sum has 2n + 2 terms. We will re-write it as two sums each having n + 1 terms.

formula

This last line represents the nth term of the expansion of sin(x + y). We now turn our attention to the right side

formula

and consider the series expansion of the term sin x cos y.

Since the series for sin x and for cos x both converge absolutely, we can write (sin x)(cos y) as the Cauchy product of the two series

formula

where

formula

and the ai, bn−i terms come from the series for sin x and cos x, respectively (Rudin 1964). Let us examine the general term cn of this Cauchy product.

formula

Then the term cn is the odd powers of x in part [1] of the general binomial expansion above. By switching x with y in the previous equation, we get the general term dn for the Cauchy product of the series for sin y and cos x.

formula

This matches the even powers of x in part [2] of the general binomial expansion.

Therefore

formula

and

formula

We now turn our attention to a special value, the smallest positive critical value of sin(x), a number we will call Q.

Theorem Critical Value There exists a smallest positive critical value of sin(x), that is, a smallest positive zero of cos(x).

Proof. We have already seen that cos0 = 1. Now observe that

formula

We now write

formula

The Remainder Theorem for alternating series tells us that

formula

and so

formula

Since cos0 > 0 and cos2 < 0, by the Intermediate Value Theorem, there is at least one real number c ∈ (0,2) with cos c = 0. The nonempty set {x | cos x = 0} is the inverse image of the closed point set {0} under the continuous function cos x. Therefore the set {x | cos x = 0} is closed. It follows that the set

formula

is nonempty, closed, bounded, and is therefore compact (Willard 1970). It must contain its least element which we shall call, temporarily, Q.

Definition of Q

formula

Consequences of the key theorems.–The Pythagorean Identity leads directly to the following corollary.

Corollary For all x,

formula

Proof: If |sin x| > 1, then cos2 x < 0 and cos x is not a real number. Similarly, if |cos x| > 1, then sin x is not a real number. In this study, we are restricting our work to real numbers.

The next two corollaries follow from the Pythagorean Identity and the special properties of Q.

Corollary sin Q = 1 and sin x has an absolute maximum value of 1 at x = Q.

Proof: Since cos0 = 1 and cos x is an even function, for x ∈ (−Q, Q), we have cos x > 0. Therefore sin x is strictly increasing on (−Q, Q). Since 0 < Q we have 0 = sin 0 < sin Q. From the Pythagorean Identity we know that

formula

Since cos Q = 0, it must be the case that sin Q = 1. We have already observed that

formula

and therefore 1 is an absolute maximum of sin x.

Corollary The range of sin x is [−1, 1].

Proof: Because sin x is an odd function we have sin(−Q) = −sin Q = −1 is an absolute minimum. The range [−1, 1] follows from the continuity of sin x and the Intermediate Value Theorem.

Later we will see that the range of cos x is also [−1, 1].

Our next two corollaries follow from the Sine Sum Theorem.

Corollary sin(xy) = sin x cos y − cos x sin y

Proof: Because sin x is an odd function and cos x is even, we have the following:

formula

Corollary sin 2x = 2 sin x cos x

Proof:

formula

We now consider the cofunction rules that follow from the Sine Sum Identity and the properties of Q. We will use these later to show that the sine and cosine functions are periodic.

Corollary Cofunction Rule sin(Qx) = cos x

Proof:

formula

Corollary Cofunction Rule cos(Qx) = sin x

Proof:

formula

In the following corollaries we complete the sum, difference, and double angle rules.

Corollary cos(x + y) = cos x cos y − sin x sin y

Proof:

formula

The following corollaries now follow.

Corollary cos(xy) = cos x cos y + sin x sin y

Proof:

formula

Corollary cos 2x = 2cos2 x − 1

Proof:

formula

We have seen that the three key theorems have led to the familiar difference formulas as well as double angle formulas. From these follow the other identities such as half-angle and product-to-sum rules. In particular, we will later need the identity

formula

Periodicity.–We will need the sine and cosine function values of 4Q to show periodicity. Here is a sequence of steps to arrive at this point.

  1. sin2Q = 2 sin Q cos Q = 2(1)(0) = 0

  2. cos2Q = sin(Q − 2Q) = sin(−Q) = −sin Q = −1.

    From this it follows that the range of cos x is [−1, 1].

  3. sin3Q = sin(Q + 2Q) = sin Q cos 2Q + cos Q sin 2Q = −1

  4. cos3Q = sin(Q − 3Q) = sin(−2Q) = −sin 2Q = 0

  5. sin4Q = 2 sin 2Q cos 2Q = 0

  6. cos4Q = sin(Q − 4Q) = sin(−3Q) = −sin(3Q) = − (−1) = 1

We now have the machinery needed to prove the periodicity of sin x and cos x.

Definition A function f (x) is periodic if there is a positive number p such that

formula

for all x. If there is a smallest positive number p for which this holds, then p is called the period of f.

Theorem Periodicity of Sine The sine function is periodic and its period is 4Q.

Proof: We first show that sine is periodic.

formula

This shows that sin x is periodic, but does not show that the period is 4Q. To show that 4Q is the period, assume, to the contrary, that there exists a number R such that 0 < 4R < 4Q and for all x,

formula

Observe that 0 < R < Q. For x ∈ (0, Q) we have cos x > 0 because cos 0 = 1 and Q is the smallest value with cos Q = 0. We also have sin x > 0 since sin0 = 0 and sin is increasing on (0, Q). Now examine sin Q:

formula

Because sin Q = 1,

formula

We now have two cases:

formula

Then by the double angle identity,

formula

If cos2 R = 1, then by the Pythagorean Identity, sin R = 0, a contradiction to the fact that sin R > 0.

formula

Then

formula

This last statement contradicts the choice of Q as the smallest positive number in [0, 2] with cos Q = 0.

Therefore such a number R does not exist, and the period of sin is 4Q.

Corollary Periodicity of Cosine The cosine function is periodic with period 4Q.

Proof: We can write cos x as

formula

Because horizontal translations and vertical rotations about the x-axis do not change the period of a function, cos x is periodic with period 4Q.

Connection to Geometry.–With this result we now show the connection between the analytic and geometric approaches to trigonometry. Figure 1 shows the area under the unit circle function from x=0 to x=1.

Theorem Connection with π

formula

Proof: Use the substitution

formula

with the values so that the integral becomes

formula

The integral represents the quarter-circle area enclosed by the unit circle, the nonnegative x-axis, and the nonnegative y-axis, and so we are led to the conclusion that

formula

Using what we have previously developed about multiples of Q, we have a table restating the values for sine and cosine in terms of π instead of Q.

From this follows the usual information about the graphs of the sine and cosine: intervals for positive/negative values, intervals for increasing/decreasing, local (and absolute) maximums/minimums.

Without geometry, we can find the values of sine and cosine of , , using only the sum and difference identities. We include the development of these values in Appendix A: Trig Functions of Special Angles (see https://doi.org/10.32011/txjsci_71_1_Article10.SO1). In Appendix B: Connection to Unit Circle Trigonometry (https://doi.org/10.32011/txjsci_71_1_Article10.SO2), we present the mathematics that connects the sine and cosine functions, defined here as power series, to the trig functions defined using the unit circle.

Pythagorean Identity revisited.–We conclude this study with the observation that the converse of the Pythagorean Identity also holds.

Theorem If f : R → R is analytic, f′(0) = 1, f (0) = 0, and f satisfies the Pythagorean Identity

formula

for all x, then f (x) ≡ sin x.

Proof: Differentiation of both sides gives

formula

so that

formula

Since f′ (0) = 1, and f is analytic, f′ is positive on some open interval containing 0. Therefore, on this interval,

formula

and f(x) = sin(x). Moreover, if two analytic functions agree on an open interval, then they agree on R.

Summary & Conclusions

We have developed the theorems and identities of basic trigonometry using the definition of the sine function as the solution, expressed as a power series, of a certain second order linear homogeneous differential equation. The key theorems in this study are the Pythagorean Identity, the Sine Sum Identity, and the special value Q, which turned out to be π/2. From these the other identities follow. The interested reader is referred to Landau, chapter 16, in which the sine and cosine functions are developed from a power series definition. In a brief note, Appendix III in Hardy uses the definition of the inverse tangent function as an integral to lead to the definitions of sine, cosine, and their sum laws.

In a future study we plan to consider a generalization of the sine and cosine functions, and show that versions of the Key Theorems still hold in these settings.

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Supplementary data