In this article, the familiar sine and cosine appear in an unexpected role - as vector entities in a multidimensional space. This is possible thanks to the unit space and its unusual properties, in which even the familiar (scalar) sine and cosine can become vector quantities. For the correct functioning of the operations of multiplication and addition of vectors, special mathematics is used here - hyperbolic vector algebra. The work presents a new approach to trigonometric functions through their vector generalization, which expands classical trigonometry and opens up new possibilities for its application.

Let's transform the familiar scalar functions into a vector according to this theorem, but we will do it separately for its cosine and sine parts. For brevity of subsequent formulas, we will denote this transformation as follows \[ \cos\v \to \Cos\v \\ \sin\v \to \Sin\v \tag{1}\] where: \(\Cos\v, \Sin\v\) - vector cosine and sine, respectively.

Let's write out the first few terms of the series of such a transformation: \[ \Cos\v = \j_0 + {1 \over \sqrt{2}} \left( \j_2 {\ik \a \over \sqrt{2!}} + \j_4 {\a^2 \over \sqrt{4!}} + \j_6 {\ik \a^3 \over \sqrt{6!}} + \ldots \right) \\ \Sin\v = {1 \over \sqrt{2}} \left( \j_2 {\a \over \sqrt{2!}} + \j_4 {\ik \a^2 \over \sqrt{4!}} + \j_6 {\a^3 \over \sqrt{6!}} + \ldots \right) \tag{2}\] Here: \(\mathbf{j_n}\) - unit vectors, \(n\) - positive integers, \(\i\) - hyperbolic unit, whose square is plus one, \(\v\) - an angle taken for convenience of display as half of \(\a\): \[\v = {\a \over 2} \] In the most general case, an equally probable plus or minus sign (\(\pm\)) is added next to each term of the series (2). For brevity, we will not write this down, but simply imply it, since for paired operations these signs will be pairwise identical and will not affect the result.

Let us also recall that when transforming a scalar into a vector, and vice versa, the equality of the squares of the scalar and vector functions is observed: \[ \Cos\v \cdot \Cos\v = \cos^2 \!\v \tag{3}\] \[ \Sin\v \cdot \Sin\v = \sin^2 \!\v \tag{4}\] It should be noted that in the scalar multiplication of two vector functions with the extension of hyperbolic numbers, in general Hermitian scalar multiplication should be applied.

Let's represent the vector cosine and sine in a simpler, and at the same time - most general form, using the following transformation. But let us write these formulas in this form: \[ \S = \sumn{1} \j_{2n}\, (\i)^n \sqrt{ \frac12 {\a^{2n} \over (2n)!} } \\ \S^{*} = \sumn{1} \j_{2n}\, (\oi)^n \sqrt{ \frac12 {\a^{2n} \over (2n)!} } \tag{5}\] \[ \Cos\v = \j_0 + \S \tag{6}\] \[ \Sin\v = \ik \S \tag{7}\] For the subsequent exposition we will need some relations, following from the rules of multiplication of hyperbolic vector functions, and Maclaurin series [1]. They will greatly reduce our calculations in the future. \[ \S^{*} \cdot \S = \sumn{1} (\oi)^n (\i)^n \frac12 {\a^{2n} \over (2n)!} \tag{8}\] Here \(\oi\) is the complex conjugate hyperbolic unit.

Since in hyperbolic algebra \[ \oi\kern1pt^n \cdot \i^n = (-1)^n \] then \[ \S^{*} \cdot \S = \sumn{1} (-1)^n \frac12 {\a^{2n} \over (2n)!} = {\cos(\a) - 1 \over 2} = -\sin^2\v \tag{9}\] It is also obvious that \[ \S \cdot \j_0 = \S^{*} \cdot \j_0 = 0 \tag{10}\] Then from expression (9) the square of the vector sine is automatically derived \[ \Sin\v \cdot \Sin\v = \oi\S^{*} \cdot \ik\S = \oi\ik\, \S^{*} \cdot \S = \sin^2\v \tag{11}\] which was presented in property (4). Property (3) can be obtained similarly.

Let's try to connect vector sine and cosine in different ways. First, let's look at the sum of their squares: \[ \Cos\v^2 + \Sin\v^2 = 1 \tag{12}\] This property is derived from formulas (3-4) and the rules of trigonometry. It should be noted that the sum of the squares of the scalar sine and cosine has a similar property.

The following product of the vector cosine and sine may seem unusual. If we follow the relations (9-10), we get \[ \Cos\v \cdot \Sin\v = (\j_0 + \S^{*}) \ik\S = -\ik\sin^2\v \tag{13}\] Multiplicationvector sine and cosine will give the same value, but with the opposite sign: \[ \Sin\v \cdot \Cos\v = \ik\sin^2\v \tag{14}\]

Now we can find the square of the sum of vector sine and cosine: \[ \left[ \Cos\v + \Sin\v \right]^2 = \Cos\v^2 + \Sin\v^2 + \Cos\v \cdot \Sin\v + \Sin\v \cdot \Cos\v = 1 \tag{15}\] In the same way we can find the square of the difference \[ \left[ \Cos\v - \Sin\v \right]^2 = 1 \tag{16}\] which will also be equal to one. This amazing property fundamentally distinguishes hyperbolic vector algebra from classical vector algebra. It can be shown that the squares of the absolute values ​​of the difference and the sum of the vector cosine and sine will also be equal to one.

But this difference of vectors can prompt us to think: \[ \Cos\v - \ik\Sin\v = \j_0 \tag{17}\] Accordingly, the absolute value of this expression will also be equal to one: \[ |\Cos\v - \ik\Sin\v| = 1 \tag{18}\]

This note presents a new understanding of the familiar sine and cosine functions — as vector objects in a multidimensional space. Using hyperbolic vector algebra, these functions acquire unusual properties, while retaining important trigonometric relations, such as the equality of the sum of their squares to one. Examples of transforming scalar functions into vector series are shown, and key identities confirming the consistency of such a system are proven. The new notation reveals additional symmetries and features that are absent in classical vector algebra.

The next article will consider products of vector sines and cosines at different angles. This will reveal new patterns of their interaction and expand the understanding of trigonometric functions in multidimensional vector form.