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Научно-методически статии
GENERALIZED TRIGONOMETRIC IDENTITIES
https://doi.org/10.53656/math2026-5-1-gti
Резюме. In this short note we present the use of a form of binary symmetry in trigonometry in order to present naturally grouped trigonometric identities in a compact way. This approach is more conceptual and, in particular, replaces a tedious consideration of similar cases.
Ключови думи: generalized trigonometric identity; binary symmetry; shortening the exposition
1. Notation, Introduction
1.1. Notation
The following notation will be frequently used in this paper:
\(\mathbb{Z}_{2}=\{0,1\}\) : the field of two elements. We also consider the external law of composition \(0 . x=0\) and \(1 . x=x\), on the set of real numbers \(x \in \mathbb{R}\) with the field \(\mathbb{Z}_{2}\) as operating set. This law satisfies the properties \((k s) . x=k .(s . x)\), \(k .(x+y)=k . x+k . y\), and \(k .(x y)=(k . x) y=x(k . y)\);
When \(-1 \in \mathbb{R}\) and \(k \in \mathbb{Z}_{2}\), the power \((-1)^{k}\) is well defined;
For \(0,1 \in \mathbb{Z}_{2}\) and any \(\alpha \in \mathbb{R}\) we denote: \(\operatorname{tr}_{0} \alpha=\cos \alpha, \operatorname{tr}_{1} \alpha=\sin \alpha\).
1.2. Introduction
An expression of the form \(I\left(k_{1}, k_{2}, \ldots, k_{n}\right)=0, k_{i} \in \mathbb{Z}_{2}\), ki ∈ Z2, where the left hand side is a polynomial with real coefficients and "unknowns" of the form \(\operatorname{tr}_{f\left(k_{1}, k_{2}, \ldots, k_{n}\right)}(\alpha, \beta, \ldots),(-1)^{g\left(k_{1}, k_{2}, \ldots, k_{n}\right)}\), (−1) g(k1,k2,...,kn) , where \(\alpha, \beta, \ldots \in \mathbb{R}\), and with polynomials \(f\left(k_{1}, k_{2}, \ldots, k_{n}\right), g\left(k_{1}, k_{2}, \ldots, k_{n}\right)\) over the f field \(\mathbb{Z}_{2}\), is said to be a generalized trigonometric identity if for any vector \(\left(k_{1}, k_{2}, \ldots, k_{n}\right)\) from \(\mathbb{Z}_{2}\)vector space \(\left(\mathbb{Z}_{2}\right)^{n}\) it is a trigonometric identity. The last statement can also be read as a hidden definition of a trigonometric identity as a specialization of a generalized one. It turns out that
(1.2.1) \[ \left(I\left(k_{1}, k_{2}, \ldots, k_{n}\right)=0\right)_{\left(k_{1}, k_{2}, \ldots, k_{n}\right) \in\left(\mathbb{Z}_{2}\right)^{n}} \]
is a family of naturally grouped trigonometric identities. For example, the generalized trigonometric identity
(1.2.2) \[ \operatorname{tr}_{k}(\theta)+(-1)^{t} \operatorname{tr}_{k}(\varphi)=2(-1)^{(k+1) t} \operatorname{tr}_{k+t}\left(\tfrac{\theta+\varphi}{2}\right) \operatorname{tr}_{t}\left(\tfrac{\theta-\varphi}{2}\right), k, t \in \mathbb{Z}_{2}, \]
produces the four well known trigonometric identities \[ \begin{gathered} k=0, t=0: \cos \theta+\cos \varphi=2 \cos \left(\tfrac{\theta+\varphi}{2}\right) \cos \left(\tfrac{\theta-\varphi}{2}\right), \\ k=0, t=1: \cos \theta-\cos \varphi=-2 \sin \left(\tfrac{\theta+\varphi}{2}\right) \sin \left(\tfrac{\theta-\varphi}{2}\right), \\ k=1, t=0: \sin \theta+\sin \varphi=2 \sin \left(\tfrac{\theta+\varphi}{2}\right) \cos \left(\tfrac{\theta-\varphi}{2}\right), \end{gathered} \] (1.2.3) \[ k=1, t=1: \sin \theta-\sin \varphi=2 \cos \left(\tfrac{\theta+\varphi}{2}\right) \sin \left(\tfrac{\theta-\varphi}{2}\right) . \] Note that the involution \((k, t) \mapsto(k+1, t)\) of the \(\mathbb{Z}_{2}\)-vector space \(\left(\mathbb{Z}_{2}\right)^{2}\) induces an involution on the above set of four trigonometric identities, which interchanges sine functions and cosine functions to the left-hand side as a mirror symmetry.
Below we present several generalized trigonometric identities and their application for solving some generalized trigonometric equations. The author uses this kind of \(\mathbb{Z}_{2}\)-symmetry in trigonometry in his paper (Iliev, 2025). This approach shortens the exposition significantly because it avoids considering of several similar cases. In Section 2 we present an example of how to construct the generalized trigonometric identity (1.2.2) which produces the family (1.2.3) of trigonometric identities.
All citations are taken from the famous book (Gelfand & Saul, 2001) and are included for the sake of completeness.
2. Construction of Generalized Trigonometric Identities
It turns out that given a family (1.2.1) of \(2^{n}\)“naturally” grouped trigonometric identities we can find their “generalized source” \(I\left(k_{1}, k_{2}, \ldots, k_{n}\right)=0\) by solving a system of linear equations over the field \(\mathbb{Z}_{2}\). As an example of this procedure we construct the generalized trigonometric identity (1.2.2), starting from the group (1.2.3) of trigonometric identities. The obvious general draft of (1.2.2) has the form
\[ \operatorname{tr}_{k}(\theta)+(-1)^{t} \operatorname{tr}_{k}(\varphi)=2(-1)^{a(k, t)} \operatorname{tr}_{b(k, t)}\left(\tfrac{\theta+\varphi}{2}\right) \operatorname{tr}_{c(k, t)}\left(\tfrac{\theta-\varphi}{2}\right), k, t \in \mathbb{Z}_{2}, \] with unknown polynomials \(a(k, t), b(k, t)\), a and \(c(k, t)\) from the ring \(\mathbb{Z}_{2}[k, t]\). Using the trial and error method, we begin with linear polynomials \(a=\) \(a_{1} k+a_{2} t, b=b_{1} k+b_{2} t\), and \(c=c_{1} k+c_{2} t\), b = b1k + b2t,and c = c1k + c2t, whose coefficients \(a_{i}, b_{i}, c_{i} \in \mathbb{Z}_{2}\) have to be determined. Taking into account (1.2.3), we obtain first that \(0=a(0,0)=0 a_{1}+0 a_{2}, 1=a(0,1)=0 a_{1}+a_{2}, 0=a(1,0)=a_{1}+0 a_{2}\), a and \(0=a(1,1)=a_{1}+a_{2}\), 1) = a1 + a2, which is a contradiction. Therefore \(a(k, t)\) cannot be linear and we try with a quadratic polynomial: \(a(k, t)=a_{1} k+a_{2} t+a_{3} k t\). We have \(0=a(0,0)=0 a_{1}+0 a_{2}+0 a_{3}, 1=a(0,1)=0 a_{1}+a_{2}+0 a_{3}, 0=\) \(a(1,0)=a_{1}+0 a_{2}+0 a_{3}\), a and \(0=a(1,1)=a_{1}+a_{2}+a_{3}\). In other words, \(a_{1}=0\), \(a_{2}=1, a_{3}=1\), a3 = 1, that is, \(a=(k+1) t\). Analogously, \(0=b(0,0)=0 b_{1}+0 b_{2}\), \(1=b(0,1)=0 b_{1}+b_{2}, 1=b(1,0)=b_{1}+0 b_{2}\), 1) = 0b1 + b2, 1 = b (1, 0) = b1 +0b2, and \(0=b(1,1)=b_{1}+b_{2}\), 1) = b1 + b2, hence \(b=k+t\). Finally, in the same way we obtain that \(c=t\).
3. Some Main Identities
3.1. Odd/Even Identities
For any \(s, t \in \mathbb{Z}_{2}\) and \(\theta \in \mathbb{R}\) one has \[ \operatorname{tr}_{s}\left((-1)^{t} \theta\right)=(-1)^{s t} \operatorname{tr}_{s}(\theta), \] see (Gelfand & Saul, 2001, Ch. 4, 5). Note that for \(s=0, t=1\) we obtain \(\cos (-\alpha)=\cos \alpha\) and for \(s=1, t=1\) we have \(\sin (-\alpha)=-\sin \alpha\).
3.2. Angle Sum and Difference Identities
For any \(s, t \in \mathbb{Z}_{2}\) and \(\alpha, \beta \in \mathbb{R}\) one has
\[ \operatorname{tr}_{s}\left(\alpha+(-1)^{t} \beta\right)=\operatorname{tr}_{s}(\alpha) \operatorname{tr}_{0}(\beta)+(-1)^{s+t+1} \operatorname{tr}_{s+1}(\alpha) \operatorname{tr}_{1}(\beta), \] see (Gelfand & Saul, 2001, Ch. 6, 2; Ch. 7, 2). Note that the specialization \(s=0, t=0\) yields the trigonometric identity \(\cos (\alpha+\beta)=\cos \alpha \cos \beta-\) \(\sin \alpha \sin \beta\), the specialization \(s=0, t=1\) yields \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\) \(\sin \alpha \sin \beta\), the specialization \(s=1, t=0\) yields \(\sin (\alpha+\beta)=\sin \alpha \cos \beta+\) \(\cos \alpha \sin \beta\), and the specialization \(s=1, t=1\) yields \(\sin (\alpha-\beta)=\sin \alpha \cos \beta-\) \(\cos \alpha \sin \beta\). Below we omit similar remarks.
In particular, for any \(s \in \mathbb{Z}_{2}\) and \(\beta \in \mathbb{R}\) one has
\[ \operatorname{tr}_{s+1}\left(\tfrac{\pi}{2}-\beta\right)=\operatorname{tr}_{s}(\beta), \operatorname{tr}_{s+1}\left(\tfrac{3 \pi}{2}-\beta\right)=-\operatorname{tr}_{s}(\beta) . \]
3.3. Product-to-Sum Identities
For any \(s, t \in \mathbb{Z}_{2}\) and \(\alpha, \beta \in \mathbb{R}\) one has
\[ \operatorname{tr}_{s} \alpha \operatorname{tr}_{t} \beta=\tfrac{1}{2}\left(\operatorname{tr}_{s+t}\left(\alpha+(-1)^{s+t+1} \beta\right)+(-1)^{t} \operatorname{tr}_{s+t}\left(\alpha+(-1)^{s+t} \beta\right)\right), \] see (Gelfand & Saul, 2001, Ch. 7, 9).
3.4. Sum-to-Product Identities
We change the variables in the identity from 3.3 by the rules \(\theta=\alpha+\) \((-1)^{s+t+1} \beta, \varphi=\alpha+(-1)^{s+t} \beta\). Substituting \(k=s+t\) and using 3.1, we obtain that for any \(k, t \in \mathbb{Z}_{2}\) and \(\theta, \varphi \in \mathbb{R}\) one has
\[ \operatorname{tr}_{k}(\theta)+(-1)^{t} \operatorname{tr}_{k}(\varphi)=2(-1)^{(k+1) t} \operatorname{tr}_{k+t}\left(\tfrac{\theta+\varphi}{2}\right) \operatorname{tr}_{t}\left(\tfrac{\theta-\varphi}{2}\right), \]
see (Gelfand & Saul, 2001, Ch. 7, 10).
4. Solutions of Some Generalized Trigonometric Equations
4.1. The Equation \(\operatorname{tr}_{k}(\theta)=0, \theta \in I\), θ ∈ I, where \(I \subset \mathbb{R}\) is a half-open interval of length \(2 \pi\)
Lemma 4.1.1 Let \(c \in \mathbb{R}\). If \(m_{c}\) is the minimal integer such that \(c+m_{c} \pi \in I\), then \(c+\left(m_{c}+1\right) \pi \in I\).
Proof: Let \(a\) a and \(b\), with \(a \lt b\), b e the endpoints of the interval \(I\). If \(a \in I\), then \(c+\left(m_{c}-1\right) \pi \lt a\) a and \(c+\left(m_{c}+1\right) \pi \lt a+2 \pi=b\). If \(a \notin I\), then \(c+\left(m_{c}-1\right) \pi \leq a\) a and \(c+\left(m_{c}+1\right) \pi \leq a+2 \pi=b\).
The solution of the equation stated in the title is
\[ \theta=(k+1) \cdot\left(\tfrac{\pi}{2}+m_{\tfrac{\pi}{2}} \pi\right)+k \cdot\left(m_{0} \pi\right)+n \cdot \pi, n \in \mathbb{Z}_{2} . \]
Remark 4.1.2 If \(I=[0,2 \pi)\) or \(I=\left[-\tfrac{\pi}{4}, \tfrac{7 \pi}{4}\right)\), then \(m_{\tfrac{\pi}{2}}=0\) and \(m_{0}=0\). If \(I=(-\pi, \pi]\) or \(I=\left[-\tfrac{3 \pi}{4}, \tfrac{5 \pi}{4}\right)\),, then \(m_{\tfrac{\pi}{2}}=-1\) and \(m_{0}=0\).
4.2. The Equation \(\operatorname{tr}_{k}(\theta)+(-1)^{t} \operatorname{tr}_{s}(\varphi)=0, \theta, \varphi \in[0,2 \pi)\)
Below we apply the identities from 3.2, 3.4, and Remark 4.1.2. Moreover, we note that \(\tfrac{\theta+\varphi}{2} \in[0,2 \pi), \tfrac{\theta-\varphi}{2} \in(-\pi, \pi]\).
Case 1. \(s=k\)
The equation \(\operatorname{tr}_{k+t}\left(\tfrac{\theta+\varphi}{2}\right)=0\) has solutions \(\tfrac{\theta+\varphi}{2}=(k+t+1) \cdot \tfrac{\pi}{2}+n \cdot \pi\), \(n \in \mathbb{Z}_{2}\), or, equivalently, \(\theta+\varphi=(k+t+1) . \pi+n .(2 \pi), n \in \mathbb{Z}_{2}\). Moreover, the equation \(\operatorname{tr}_{t}\left(\tfrac{\theta-\varphi}{2}\right)=0\) has solutions \(\tfrac{\theta-\varphi}{2}=(t+1) .\left(-\tfrac{\pi}{2}\right)+n . \pi, n \in \mathbb{Z}_{2}\), or, equivalently, \(\theta-\varphi=-(t+1) . \pi+n .(2 \pi), n \in \mathbb{Z}_{2}\).
Case 2. \(s=k+1\)
We obtain
\[ \begin{aligned} & \operatorname{tr}_{k}(\theta)+(-1)^{t} \operatorname{tr}_{k+1}(\varphi)=\operatorname{tr}_{k}(\theta)+(-1)^{t} \operatorname{tr}_{k}\left(\tfrac{\pi}{2}-\varphi\right)= \\ & \quad 2(-1)^{(k+1) t} \operatorname{tr}_{k+t}\left(\tfrac{\theta-\varphi}{2}+\tfrac{\pi}{4}\right) \operatorname{tr}_{t}\left(\tfrac{\theta+\varphi}{2}-\tfrac{\pi}{4}\right) \end{aligned} \] Moreover, \(\tfrac{\theta+\varphi}{2}-\tfrac{\pi}{4} \in\left[-\tfrac{\pi}{4}, \tfrac{7 \pi}{4}\right), \tfrac{\theta-\varphi}{2}+\tfrac{\pi}{4} \in\left(-\tfrac{3 \pi}{4}, \tfrac{5 \pi}{4}\right]\).
The equation \(\operatorname{tr}_{k+t}\left(\tfrac{\theta-\varphi}{2}+\tfrac{\pi}{4}\right)=0\) has solutions \(\tfrac{\theta-\varphi}{2}+\tfrac{\pi}{4}=(k+t+\) \(1) .\left(-\tfrac{\pi}{2}\right)+n . \pi, n \in \mathbb{Z}_{2}\), or, equivalently, \(\theta-\varphi=-(k+t+1) . \pi+n .(2 \pi)-\tfrac{\pi}{2}\), \(n \in \mathbb{Z}_{2}\).
The equation \(\operatorname{tr}_{t}\left(\tfrac{\theta+\varphi}{2}-\tfrac{\pi}{4}\right)=0\) has solutions \(\tfrac{\theta+\varphi}{2}-\tfrac{\pi}{4}=(t+1) \cdot \tfrac{\pi}{2}+n \cdot \pi\), \(n \in \mathbb{Z}_{2}\), or, equivalently, \(\theta+\varphi=(t+1) . \pi+n .(2 \pi)+\tfrac{\pi}{2}, n \in \mathbb{Z}_{2}\).
Acknowledgements
It is a pleasure for me to cordially thank Prof. D.Sc. Nikolai Nikolov, Corresponding member of BAS, who read a preliminary draft of this paper and suggested me to submit it for publication in this magazine. I would also like to express my sincere thanks to the referees for their very useful remarks which are invaluable for making this paper more readable. Last but not least, I would like to thank administration of the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences for creating a safe working environment.
REFERENCES
Gelfand, I. M. & Saul, M. E. (2001). Trigonometry. Birkhäuser.
Iliev, V. V. (2025). On the Protection Against Noise for Measurement-Based Quantum Computation. Reliability: Theory & Applications, 20(2), 324 – 331.