Trigonometric Ratios Using Algebraic Methods

The main aim of this article is to start with an expository introduction to the trigonometric ratios and then proceed to the latest results in the ﬁeld. H istorically, the exact ratios were obtained using geometric constructions. The geometric methods have their own limitations arising from certain theorems. In view of the certain limitations of the geometric methods, we shall focus on the powerful techniques of equations in deriving the exact trigonometric ratios using surds. The cubic and higher-order equations naturally arise while deriving the exact trigonometric ratios. These equations are best expressed using the expansions of the cosines and sine of multiple angles using the Chebyshev polynomials of the ﬁrst and second kind r espectively. So, we brieﬂy present the essential properties of the Chebyshev polynomials. The equations lead to the question of reduced polynomials. This question of the reduced polynomials is addressed using the Euler’s totient function. So, we describe the techniques from theory of equations and reduced polynomials. The trigonometric ratios of certain rational angles (when measured in degrees) give rise to rational trigonometric ratios. We shall discus s these along with the related theorems. This is a frontline area of research connecting trigonometry and number theory. Results from number theory and theory of equations are presented wherever required.


Introduction
The exact values of the trigonometric ratios have been a subject of keen interest since the beginning trigonometry and continues to this day [1]- [7]. The Greeks relied on geometric meth-ods to obtain the trigonometric ratios. The same tradition was followed by the Medieval Arab scientists and scientists from India and China [8]. Some of these historic aspects were covered during the International Year of Light and Light-based Technologies [9]- [15]. Geometric methods are constrained by the constructible regular polygons [16]. This necessitates the use of equations, which enables us to obtain the trigonometric ratios of angles not possible from the geometric constructions. "A trigonometric number is an irrational number obtained by taking the sine or cosine of a rational number of degrees (if expressed in radians, the angles is a rational multiple of π)." The only exceptions are cos α , sin α ∈ {0, ± 1 2 , ±1}. We shall look at the theorems related to the rational and irrational trigonometric numbers.
For completeness, we briefly cover the procedures for obtaining the exact values of the trigonometric ratios of the routine angles (15 • , 18 • , 30 • , 36 • , 45 • , 54 • , 60 • , 72 • and 75 • ). In Section 2, it will be seen that the basic trigonometric identities along with quadratic equations are sufficient for such derivations. In Section 3, we shall focus on the rational angles in degrees. In Section 4, using cubic equations, we obtain some additional results. The same Section has some general results from the theory of equations and number theory pointing to the limitations inherent in the geometric techniques. Consequently, there is a dependence on higher-order equations in deriving the exact ratios. Section 5, our final Section has the concluding remarks. The study of the trigonometric ratios is an ongoing topic of active research, which is poined by the continued journal articles.

Trigonometric Ratios from Identities and Equations
The very basic identity, sin 2 A + cos 2 A = 1 is a consequence of the Pythagorean theorem. The identities for the sums 900 Trigonometric Ratios Using Algebraic Methods of two angles are Matrices enable us to write them in a compact form A special case is These formulae can be derived using various techniques such as the de Moivre's formula. Alternately, we can use Euler's formula with complex arguments The identities in (1) The identities involving the difference of two angles are For the multiple angles, we have and The identities in (8) can be written as If we choose A = 45 • , then Starting with the exact value of cos A and the repeated use of the identities in (9) enables us to obtain the exact ratios of the sub-multiples of A using only square-roots [17]. On taking k steps, that is k square-roots, we reach the exact value of sin(A/2 k ) expressed as the k-th root The same procedure leads to an exact expression for cos(A/2 k ) 2 cos A = 2 + 2 cos(2A) = 2 + 2 + 2 cos(2 2 A) = 2 + 2 + 2 + 2 cos(2 3 A) = 2 + 2 + 2 + 2 + 2 cos(2 4 A) Many exact values can be obtained by a suitable combination of the various identities. Based on the combinations of the identities, the same trigonometric ratio can have several equivalent forms as seen in the following example where φ = ( √ 5 + 1)/2 is the celebrated golden ratio [18]. Identities in (1) enableus to obtain These identities are sufficient to obtain the trigonometric ratios of This leads to the equation whose solution is The basic identity readily gives the required value of cos 18 • = sin 72 • Using the identities in (8), the trigonometric ratios of 36 • and 54 • are In this Section, we relied on trigonometric identities along with quadratic equations. In Section 4, we shall see that we require not only cubic or quartic but even higher order equations.

Irrationality of Trigonometric Ratios
In 1956, Ivan Morton Niven published a book [19,20], which had the following theorem summarizing the results on the irrationality of the trigonometric ratios Theorem 1 Niven's Theorem: The only rational values of α in the interval 0 • ≤ α ≤ 90 • for which the sine of α degrees is also a rational number are This theorem restricted the rational values to the sets, cos α , sin α ∈ {0 , ± 1 2 , ±1}, sec α , csc α ∈ {±1, ±2} and tan α , cot α ∈ {0 , ±1}. The arithmetic properties of trigonometric functions have been a recurring topic in the mathematical literature. Part of the aforementioned results were known as early as 1922 with several independent contributions: Swift (1922 [21]), Underwood (1921 [22]), Lehmer (1933 [23]), Olmsted (1945 [24]), and later by Niven (1956) who stated it in the above form. It came to be known as Niven's Theorem [19,20]. Later on, there appeared several proofs to this important theorem employing a variety of techniques such as induction, de Moivre formulas, Chebyshev polynomials, cyclotomic polynomials among others (see [25]- [35] and references therein).
The proof is based on the periodicity of the cosine function [35]. Irrational numbers can be learned through basic trigonometry [37]. The trigonometric functions can be defined as solutions of differential equations. Hence, the irrationality of the values trigonometric functions can be deduced using techniques from calculus [37]- [39].
The irrationality of the trigonometric ratios has found applications within mathematics and several areas of physical sciences [40]- [41]. As an example, we note the Gregory numbers defined as where x is any integer or any rational number. For instance, with x = 1, G 1 = tan −1 (1) = 45 • is a Gregory number. Apart from the two exceptional case, G −1 = −45 • and G 1 = 45 • , every Gregory number when expressed in degrees is an irrational number. Historically, the identities involving the arctangent function have been widely used to compute the values of π to a very large number of decimal places [27]. The following theorem by Arno Berger [33] summarises the rational linear independence of trigonometric numbers Theorem 4 Berger Theorem: Let r 1 and r 2 to be two rational numbers such that either r 1 − r 2 and r 1 + r 2 is not an integer, then the three numbers 1, cos(r 1 π) and cos(r 2 π) are rationally independent.
covered in the previous sections. The idea is to derive the expression for cos(nx) and sin(nx) as polynomials of cos x and sin x. This is best done using the Chebyshev polynomials of first kind and second kind respectively [5,6,7,42].

Expansion of cos(nθ)
Let us first examine the pattern of cos(nθ) for the first few values of n by the repetitive use of the identities in (1) cos(2θ) = 2 cos 2 θ − 1 , The identities in (21)-(24) suggest that cos(nθ) is a n-th degree polynomial in cos θ. This is indeed the case. We note the general result where T n (x) denotes the Chebyshev polynomials of the first kind [5,6,7,42,43]. The first few are Chebyshev polynomials of the first kind satisfy the recurrence relation The leading coefficient of each T n (x) is 2 n−1 . In the context of solving polynomial equations, we have the following observations about T n (x). The even-order T n (x) are function of x 2 , leading to equations of degree n/2 in terms of polynomials of

Expansion of sin(nθ)
Now, we look for the pattern of sin(nθ) for the first few values of n sin(2θ) = 2 sin θ cos θ = sin θ {2 cos θ} , (29) sin(3θ) = 3 sin θ − 4 sin 3 θ , sin(4θ) = 4 sin θ cos θ 2 cos 2 θ − 1 , sin(5θ) = 5 sin θ − 20 sin 3 θ + 16 sin 5 θ , = sin θ 16 cos 4 θ − 12 cos 2 θ + 1 . (32) The identities in (29)- (32) suggest that sin(nθ) is sin θ times a (n − 1)-th degree polynomial in cos θ. We note the general result sin(nθ) = sin θ U n−1 (cos θ) , where U n (x) denotes the Chebyshev polynomials of the second kind [5,6,7,42,43]. The first few are Chebyshev polynomials of the second kind satisfy the recurrence relation The leading coefficient of each U n (x) is 2 n . In the context of solving polynomial equations, we have the following observations about U n (x). The even-order U n (x) are function of x 2 , leading to equations of degree n/2 in terms of polynomials of x 2 . The odd-order U n (x) are functions of the odd powers of x and do not have the constant term. So, the odd-order T n (x) can be factorised into x times a polynomial of degree (n − 1)/2 in x 2 . The polynomials, U n (x) can also be obtained from the following n × n determinant equation .

General Cubic Equation Formula
The most general cubic equation is The three solutions are expressed through the following where ω is the primitive cube root of unity from the equation The three roots are In the present context of trigonometric equations, we are dealing with cubic and higher order equations with real coefficients (mostly integer coefficients). We also know that the relevant trigonometric equations do not have rational solutions. From the Galois theory, we know that when none of the roots are rational and when all three roots are distinct and real, it is impossible to express the roots using only real radicals.

Trigonometric Ratios of 20 • (π/9)
Let us now consider the case of 20 • . In (30), we choose θ = 20 • , with x = sin 20 • and obtain the cubic equation The solution for sin 20 • is In (22), we choose θ = 20 • , with x = cos 20 • and obtain the cubic equation The solution for cos 20 • is

Trigonometric Ratios of 1 • (π/180)
The exact value of sin 1 • can be calculates from sin 3 • through the cubic equation. Let x = sin 1 • , then The solution of this equation is Some additional but equivalent expressions for sin 1 • are available in [44].

Minimal Polynomials
If the coefficients of a polynomial equation are rational, then the denominators of the coefficients can be removed. This results in an equivalent polynomial with only integer coefficients. As noted, the Chebyshev polynomials have only integer coefficients and their solutions leads to the values of the trigonometric functions. In Section 3, we noted that cos(2π/n) is a rational number if n = 1, 2, 3, 4, and 6. This implies that for these values of n, cos θ satisfies a linear equation with integer coefficients. We also noted that cos(2π/n) is obtained from quadratic equations with integer coefficients for n = 5, 8 and 12. The algebraic numbers are roots of polynomials with integer coefficients. If a real (or complex) number is a root of an irreducible polynomial of degree n with integer coefficients, it is said to be an algebraic number with algebraic degree n. The corresponding irreducible polynomial is its minimal polynomial. Here, irreducible polynomial means that it cannot be factored into lower degree polynomials with integer coefficients.
As an example, we will determine the value of cos 30 • (i.e., cos(π/6)), using the Chebyshev polynomials. The corresponding polynomial with θ = π/6 and x = cos(π/6) is T 6 (x) = 32x 6 − 48x 4 + 18x 2 − 1. Then, the equation is We recover the familiar solution cos 30 • = √ 3/2. It is the second degree polynomial (4x 2 − 3), which leads to the value of x = cos(π/6) and is called as the reduced polynomial. Consequently, cos 30 • is an algebraic number of degree two. In the case of cos(20 • ) (i.e., cos(2π/18)), the corresponding polynomial with x = cos(2π/18) is T 18 (x). This is a polynomial of degree eighteen. But we know that the cos(20 • ) satisfies a cubic equation in (44) and is an algebraic number of degree three. This leads us to the question of the reduced polynomials and their degrees. The Chebyshev polynomials need not be the reduced polynomials all the time! This is evident from the previous two examples of cos(π/6) and cos(2π/18) which were reduced from 6 to 2 and 18 to 3 respectively.

Quartic and Higher-Order Equations
Like the cubic equation, there is a formula for the quartic equation. But for fifth and higher degree equations, there is no such formula. In fact, there is a theorem which states that such formulae do not exist! Theorem 5 Abel-Ruffini Theorem: In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions.
This theorem is also known as the Abel's impossibility theorem. The proof of this negative theorem is based on the Galois theory. Galois theory has been able to prove the impossibility of several problems of antiquity including: (a) Trisecting the angle (using only a compass and a straightedge); (b) Squaring the circle (i.e., constructing a square of area equal to the area of a given circle); (c) Doubling the cube, (i.e., constructing a cube with twice the volume of a given cube); and (d) constructing a heptagon.

Concluding Remarks
The trigonometric functions occur across mathematics and sciences. Obtaining exact values has attracted lot of attention since the beginning of the subject. Geometric techniques provide exact values in the case of several angles but are not suitable for other angles. There are several theorems which limit the use of geometric techniques in finding the exact values. The derivation of the exact trigonometric values is intimately tied to several areas such as theory of equations, number theory 905 and algebraic geometry. The irrational sets of trigonometric ratios of rational angles were covered along with related theorems. The new theorems extending the classical results were also covered. Results from number theory and theory of equations were presented wherever required. Trigonometric functions also arise in the study of the hypersingular integral equations [61] and the superstability solutions of The pexiderized trigonometric functional equations [62] Some of the numerical results can also be derived using the Microsoft Excel [63]- [68]. The other alternatives are the symbolic packages, such as the Mathematica [69,70]. MS Excel is valuable for certain types of numerical analysis [64]- [68]. It has been useful in numerous applications such as the analysis of quadratic surfaces [71]- [74]; networks of equal resistors [75]- [78]; chemical physics [79]; and theory of numbers [80,81].