Extended Angle Sum and Di ﬀ erence Identity Rules for Scalene Triangles

,


Introduction
The discovery of the identity rules is intimately tied to the development of Trigonometry -a branch of mathematics that studies the relationships between lengths and angles of sides of triangles. The etymology of the word trigonometry is Trigonon (triangle) plus Metron (measure), both expressions originating from Latin derivatives of Greek words (Johnson, 2016). The angle sum and difference identity rules essentially establish an arithmetic expression between the normalized sides (i.e. expressed in terms of sines and cosine functions) of two triangles superimposed in such a way that their individual reference angles A and B add to form the reference angle A + B of the larger triangle resulting from the superposition. The opposite is true when the angles are subtracted. Ptolomy theorem offers probably one of the oldest and most well-known proof of the angle sum and difference formulas for sines and cosines (Joyce 2013). The identity rules are a mathematical cornerstone in the Canadian educational system (Canadian Ministry of Education, 2020), making this subject of interest to students and professionals alike. Other popular proofs are also widely available (Ren 1999, Kung 2008, Smiley 2018, Smiley et al 2018, some presenting all six trigonometric angle sum and difference identities in one drawing (Nelsen 2000). Research in this topic is still ongoing in modern mathematics, with new geometrical developments of these trigonometric angle sum and difference identities being presented today (Ollerton 2018). For a right-angled triangle, the angle sum and difference identity rules for α = A ± B are sin(A ± B) = sin(A) cos(B) ± cos(A) sin (B) (1) cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B) All existing proofs share one common restrictive assumption -they all apply only to right-angle triangles comprising of one variable reference angle α and one fixed obtuse angle of γ = π/2, invoking inherently the traditional forms of sin(α) and cos(α) functions to establish a relation between the sides [governed by the Pythagoras theorem sin 2 (α) + cos 2 (α) = 1]. On the other hand, the sides of a scalene triangle ( Figure 1) are interrelated by the extended expressions for the sin * (α, γ) and cos * (α, γ) that were proved to be sin * (α, γ) = sin(α) sin(γ) ; cos * (α, γ) = cos(α) + cos(γ) sin(γ) sin(α) = sin(α + γ) sin(γ) satisfying the more general governing equation that is the Law of Cosines sin * (α, γ) 2 − 2 sin * (α, γ) cos * (α, γ) cos(γ) + cos * (α, γ) 2 = 1 (of which the Pythagoras theorem is a particular case with γ = π/2). From this outcome, the inher-ent question to ask is, how will the angle sum and difference identity rules look like when applied to scalene triangles (governed by reference angle α = A ± B and obtuse angle γ)? Figure 1. Extending the applicability of sine and cosine functions to scalene triangles [1].

Hypothesis
Just as there are angle sum and difference identity rules for sine and cosine functions that govern (the particular case of) right-angled triangles, there must exist an equivalent set of identity rules for the extended sine and cosines functions governing (the general case of) scalene triangles, which should naturally be derived from evolutive modifications into the existing proofs of angle sum and difference identity rules.

Theory
Start with geometrical relations to define mathematical equations. Prove that they reduce back to the original form. First this is done for the sum of angles, and second for the difference between angles.

Extended Angle Sum Identity Rule
Theorem 1 (Angle Sum Identity Rule for Scalene Triangles). If the extended sin * (α, γ) and cos * (α, γ) functions are the normalized projected side lengths of a scalene triangle -where angle α is the reference angle formed between the unit side and the projected extended diagonal or vertical sine side of the triangle, and the angle γ is the obtuse angle formed between the two projected sine and cosine sides of the triangle -then, for the particular case where the reference angle α is the result of the sum between smaller angles A and B such that α = A + B, the relationship between lengths given for this particular case by the extended sine and cosine are given as Proof. In addition to proving Theorem 1, it will also be shown that the extended functions sin * (α, γ) and cos * (α, γ) given by Eqs.(4-5) reduce back, for the particular case of a right-angled triangle (where γ = π/2), to the original form and sin(α) given cos(α) by Eqs.
(3). Let us start by formulating the extended sine function sin * (A + B, γ) from first principles.
Consider the scalene triangle ADB in Figure 2 that presents a reference angle ∠BAD ≡ α = A and an obtuse angle ∠ADB ≡ γ. A second scalene triangle ABC is also seen on top of the first ADB having as a reference angle CAB ≡ α = B and obtuse angle ∠ABC ≡ γ. Here, the side AB of triangle ABC equals the longest side of triangle ADB. Together, the triangles ABC and ADB define the larger triangle AFC, which has a reference angle ∠CAF ≡ α = A + B and obtuse angle ∠AFC ≡ γ. The longest side AC of triangle AFC has a unit length. Overall, all the diagrams in this article were drawn with the open-source software Geogebra (Feng 2014). Concerning triangle ABC, the side adjacent to α = B is AB = cos * (B, γ), while its opposing side is BC = sin * (B, α). For triangle ADB, the side adjacent to α = A is AD = cos * (B, γ) cos * (A, γ), while its opposing side is An important triangle formed by the interference between ADB and ABC is that of CEB at the top right corner, which possesses an internal reference angle ∠ECB ≡ A and opposing angle between CE and EB of ∠CEB ≡ ∠ ≡ π − γ. For triangle ABC, the side opposing angle α = B is CB = sin * (B, γ). This means that the triangle CEB, which is defined by angles α = A and π − γ, the length of the side CE adjacent to angle α = A is Now, the projection of the longest side AC of the triangle AFC opposing angle α = A + B is This means that when replacing the aforementioned expressions for CF [in Eq. (8)], CE [in Eq. (7)] and EF = BD [in Figure 2 to the right], while at the same time re-arranging, the resulting expression for the extended angle sum identity rule for sine -previously defined in Eq.(4) -is given as where the extended functions sin * (α, γ) and cos * (α, γ) are defined in Eq.
(3). This completes the first part of the proof. It will now be shown that Eq.(9) reduces back to the already proven form of Eq.(3) with the reference angle being replaced directly with α = A + B, or Start by defining the extended sine function sin * (α, γ) for each of the two angles A and B At the same time, define the extended cosine function cos * (α, γ) for angle B Both Eq.(11) and Eq.(12) will be replaced in Eq.(9) at a later stage. For triangle CEB, the extended cosine function for angle reference A and obtuse angle is π − γ has the expression where the angle difference identities between π and γ for sine and cosine are given by the following relations Substituting these in Eq.(13) simplifies to Replacing Eq.(11), Eq. (12) and Eq. (16) into Eq. (9) gives Expanding the terms between brackets And re-arranging, results in The second term between brackets vanishes, reducing Eq. (17) to Knowing that the angle sum identity for sine is , further simplifies this expression to which is by definition the extended sine function sin * (α, γ) with α = A + B [as stated in Eq. (10)]. The reduction of Eq.(9) to Eq.(21) -a particular case of Eq.
(3) that already proved to be true -implies that Eq.(9) is also inherently true. This completes the first part of the proof concerning the extended sine function only.
Let us now advance to the extended cosine expression cos * (A+ B, γ) and in formulating its expression from first principles. From Figure 2, the triangle CEB has a side opposing angle α = A equal to EB = FD = sin * (B, γ) sin * (A, π − γ). The projection of the longest side AC of the triangle AFC adjacent to angle α = A + B is AF = cos * (A + B, γ), which is also the difference AF = AD − FD. Replacing the aforementioned expressions for AD [in Figure 2 at the bottom] and FD [in Figure 2 to the right], while re-arranging, results in the required Eq. (5) [here conveniently renumbered as Eq. (22)] The extended cosine functions cos * (A, γ) and cos * (B, γ), and the extended sine function sin * (B, γ), are defined in Eq. (12) and Eq.(11), respectively. The remainder unknown term is the extended sine function sin * (A, π − γ), which is defined by the triangle CEB whose obtuse angle is π − γ (instead of the frequent γ), resulting in Substituting Eq.(11), Eq. (12) and Eq.(23) into Eq. (22) gives Expanding the terms betwen brackets (3) already proven to be true -implies that Eq. (22) is also inherently true. This completes the proof.

Extended Angle Difference Identity Rule
Theorem 2 Angle Difference Identity Rule for Scalene Triangles). If the extended sin * (α, γ) and cos * (α, γ) functions are the normalized projected side lengths of a scalene triangle -where angle α is the reference angle formed between the unit side and the projected extended diagonal or vertical sine side of the triangle, and the angle γ is the obtuse angle formed between the two projected sine and cosine sides of the triangle -then, for the particular case where the reference angle is the result of the difference between smaller angles A and B such that α = A − B, the relationship between lengths given for this particular case by the extended sine and cosine are given as Proof. Let us start with the extended sine function sin * (A − B, γ). Consider the scalene triangle ABD of reference angle ∠BAD ≡ α = A and obtuse angle ∠ADB ≡ γ in Figure 3. As seen before in Figure 2, the second scalene triangle ABC of reference angle ∠BAC ≡ α = B and obtuse angle ∠ABC ≡ γ is formed on top of the first -this time inverted -such that its smaller length AB equals the longest length of tringle ABD.
That is, the new triangle is the same as in Figure 2, except that it is mirrored about length AB. Together, their subtraction defines the smaller triangle AFC of reference angle ∠CAF ≡ α = A − B and obtuse angle ∠AFC ≡ γ. The important difference from Figure 2 is that the internal angles of the triangle BEC have changed with the inversion, becoming = A − π + 2γ for the internal reference angle ∠CBE and π − γ for the obtuse angle ∠BEC. The term obtuse is used here to identify the angle playing the role of π/2 in a right-angled triangle. For a scalene triangle, this will change and will not always be literally the case, but once the proof is complete it will show that Eq.(29) and Eq.(30) are both true for any combination of α, γ ∈ . In Figure 3, the oblique projection of the longest side AC of the triangle AFC is CF = sin * (A − B, γ), which is also the difference CF = BD − BE.
We will now prove that Eq.(31) holds true for any reference angle α = A − B and obtuse angle γ where α, γ ∈ . The process is the same as before in section 3.1, except for the change of sign between the two products in Eq.(9) and the extended cosine term in the second product that changes from cos * (A, π − γ) to cos * (A − π + 2γ, π − γ), as discussed due to the changes in the angles of triangle BEC. This modified cosine term is first expanded using Eq.
(3) to Replacing expressions for sin(π − γ) from Eq. (14) and for cos(π − γ) from Eq. (15) gives The terms sin(A − π + 2γ) and cos(A − π + 2γ) are further elaborated -where the angle α = A − π + 2γ is conveniently re-written as the difference between two angles -by the traditional angle sum and difference identity rules [given by Eq.
(3)] with α = A − B. As before, the reduction of Eq.(59) to Eq.(74) -a particular case of Eq.(12), already shown to be true -implies that Eq.(22) is also inherently true. This completes the proof.