trigonometric functions by using the imaginary number. Key words: Euler's formula, Moivre's formula, imaginary number, trigonometric functions. Introduction.
Presentation of Euler's formula using the imaginary number Siniša Vlajić Department of software engineering, Faculty of organisational sciences, University of Belgrade Abstract: This paper discuss the well-known Euler's formula and one possible way of its representation by the imaginary number. In the derivation of new formula which connects Euler's formula and the imaginary number is used Moivre's formula. New-obtained formula enables also presentation of sine and cosine trigonometric functions by using the imaginary number. Key words: Euler's formula, Moivre's formula, imaginary number, trigonometric functions
Introduction Euler's formula [1], as is well known, has a major role in mathematics, physics and engineering. This formula establishes the relationship between the trigonometric functions and the complex exponential function: xi
e^ = cos(x) + i sin(x),
f1
where x is any real number (given in radians) and it is argument of functions cos and sin. These functions represent trigonometric functions cosine and sine respectively. The number e is the base of the natural logarithm and i is the imaginary unit. In this paper, new formula is obtained from the Euler's formula and it establishes the relationship between the Euler's formula and the imaginary number. The derivation is performed by using De Moivre's formula [2]: (cos(x) + i sin(x))^y = cos(xy) + i sin(xy),
f2
where are x ∈ R, y ∈ Z. Subsequently, the trigonometric functions sine and cosine also are presented by using the imaginary number.
Presentation of Euler's formula using the imaginary number If x in the formula f1 is replaced by π/2 and the left and right sides of f1 raised to the power of y, where y ∈ Z, the formula obtained the following form: e^
(π/2)iy
= cos (π/2)y + i sin (π/2)y
f3
If the imaginary number i raised to the power of y, then it is possible make the relationship among Euler's formula and the imaginary number: e^
(π/2)iy
= cos (π/2)y + i sin (π/2)y = i^
y
f4
It can be presented by Table 1. Table 1: The relationship between Euler's formula and the imaginary number y
y 1 2 3 4 5 6
i^ i -1 -i 1 i -1
...
...
cos (π/2)y 0 -1 0 1 0 -1 ...
i sin (π/2)y i 0 -i 0 i 0 ...
cos (π/2)y + i sin (π/2)y i -1 -i 1 i -1 ...
1
Presentation of sine and cosine trigonometric functions using the imaginary number If the left and right sides of formula f5: cos (π/2)y + i sin (π/2)y = i^
y
f5
raised to the power of 2, it follows that : 2 2 2y cos^ (π/2)y + 2 cos (π/2)y * isin(π/2)y - sin^ (π/2)y = i^ 2
f6
2
If in the formula f6, sin^ (π/2)y is replaced by 1- cos^ (π/2)y, then is: 2 2y 2cos^ (π/2)y + 2 cos (π/2)y * isin(π/2)y – 1= i^
f7
The formula f7 can be written: 2y 2cos (π/2)y (cos(π/2)y + isin(π/2)y ) = i^ + 1 After simplification the formula f7 is: y 2y 2cos (π/2)y * i^ =i^ + 1 The cosine function can be represented by using imaginare number: 2y y cos (π/2)y = (i^ + 1)/(2* i^ )
f8
It can be presented by Table 2. Table 2: The relationship between cosine function and the imaginary number
y 1 2 3 4 5 6
cos (π/2)y 0 -1 0 1 0 -1
...
(i^
2y
...
y
+ 1)/(2* i^ ) 0 -1 0 1 0 -1 ...
The sine function can be represented by using imaginare number, where is sqr()function of square root: sin (π/2)y = sqr((i^
2y
- 1)/(2* i^ 2y))
f9
It can be presented by Table 3. Table 3: The relationship between sine function and the imaginary number
y 1 2 3 4 5 6 ...
sin (π/2)y 1 0 -1 0 1 0 ...
sqr((i^
2y
- 1)/(2* i^ 2y)) -1 or 1 0 -1 or 1 0 -1 or 1 0 ...
Conclusion This paper provides a new angle of the observation of relationships between Euler's formula, sine and cosine trigonometric functions and the imaginary number. In this way, we make the good basis for a deeper analysis and understanding of the imaginary number and its relations to the trigonometric functions and mathematical constants e, which will be the subject of further research.
References 1. Hazewinkel, Michiel, ed. (2001), "Euler formulas", Encyclopedia of mathematics, Springer, ISBN 9781-55608-010-4 2. Lial, Margaret L.; Hornsby, John; Schneider, David I.; Callie J., Daniels (2008). College Algebra and Trigonometry (4th ed.). Boston: Pearson/Addison Wesley. p. 792. ISBN 9780321497444.
2
