Trigonometria e números complexos: uma abordagem elementar com aplicações

Abstract

The present paper presents an elementary approach to trigonometry and complex numbers so that students at the end of elementary school or early high school can begin their studies in the subjects in question. The metric relations are defined in the right-angled triangle so that the Pythagorean Theorem is established. The elementary trigonometric functions sine, cosine, and tangent are introduced in the context of the acute angles via trigonometric ratios in the right-angled triangle. The Laws of the Sine and the Cosine are then established. Then the trigonometric cycle is introduced, establishing the relation between real numbers and arcs in the cycle in order to extend the elementary trigonometric functions as well as to extend the scope of the Fundamental Relationship of Trigonometry. The periodic character of these functions appears, and the notion of reducing an arc to the first quadrant is established so that the parity of these functions is stable. The sine an consine formulas of arcs sum are established and with them are still established similar result lengths for acute angles. Finally, the complex numbers are presented highlighting its trigometric representation and the De Moivre Laws. Applications of the theory are exposed to evidence its importance and scope.