Trigonometry: SINE, COS, AND TG - Review

Introduction

Relevance of the Theme

Trigonometry is one of the fundamental and central disciplines of Mathematics, with applications in various areas such as Physics, Engineering, and Computer Science. The study of sine, cosine, and tangent (sin, cos, tan) is essential to understand the relationships between angles and sides of triangles, allowing, for example, to calculate the inaccessible distance from a point, find the position of an object in space, and model waves and oscillations. Such applications make this topic extremely relevant and reinforce the need for a thorough review.

Contextualization

Trigonometry is generally studied after Plane Geometry and precedes the study of Geometry in Space. It provides a valuable set of mathematical tools for calculating inaccessible measurements and modeling natural phenomena. It is part of the core of Mathematics in the 3rd year of High School, with content that interconnects with Geometry, Algebra, and Differential and Integral Calculus. Therefore, reviewing the concepts of Sine, Cosine, and Tangent is a crucial step for a solid understanding and mastery of these tools.

Theoretical Development

Components

• Sine (sin): In a right triangle, the Sine of an acute angle is defined as the ratio between the length of the side opposite that angle and the length of the hypotenuse.

  • Opposite Side: The line segment drawn perpendicular to one of the sides of the triangle, which is not the hypotenuse, and has one of its ends at the vertex of the angle being considered.
  • Hypotenuse: The longest side of the triangle, opposite the right angle.

• Cosine (cos): In a right triangle, the Cosine of an acute angle is defined as the ratio between the length of the side adjacent to that angle and the length of the hypotenuse.

  • Adjacent Side: The line segment that connects one of the acute angles to the opposite vertex, and is not the hypotenuse.

• Tangent (tan): In a right triangle, the Tangent of an acute angle is defined as the ratio between the length of the side opposite that angle and the length of the side adjacent to that angle.

Key Terms

• Acute Angle: It is any angle that measures less than 90 degrees, that is, it is located inside a right triangle.

• Right Triangle: A triangle that has a right angle, that is, an angle of 90 degrees.

Examples and Cases

1. Example of Calculation of Sine, Cosine, and Tangent: Consider a right triangle with an angle of 30 degrees. The sine of this angle is equal to half the hypotenuse (assuming the hypotenuse is 2 units) - putting into practice the definition of Sine, sin(30 degrees) = opposite side / hypotenuse = 1/2 = 0.5. For the Cosine, it is equal to half the square root of 3, cos(30 degrees) = adjacent side / hypotenuse = sqrt(3)/2 ≈ 0.866. And for the Tangent, it is equal to the square root of 3 divided by 3, tan(30 degrees) = opposite side / adjacent side = 1 / sqrt(3) = sqrt(3) / 3 ≈ 0.577.

2. Application of the Fundamental Identity of Trigonometry: The fundamental identity sin^2(θ) + cos^2(θ) = 1 is widely used in trigonometry. In a proof using the identity, one can verify if it is valid for a certain value of θ.

Detailed Summary

Relevant Points

• Definition of Sine, Cosine, and Tangent: In a right triangle, the Sine (sin) of an acute angle is the ratio between the opposite side and the hypotenuse. The Cosine (cos) is the ratio between the adjacent side and the hypotenuse. Finally, the Tangent (tan) is the ratio between the opposite side and the adjacent side.

• Understanding the Components of a Right Triangle: In trigonometry, it is essential to understand that sine, cosine, and tangent depend on the measurements of the right triangle, more specifically on the legs and the hypotenuse.

• Interpretation of Acute Angles and Right Blades: The theory of acute angles and right blades is one of the pillars that support the study of trigonometry. Definitions and understanding involving these concepts are vital for the full understanding of sines, cosines, and tangents.

Conclusions

• Practical Applicability of Trigonometry: Trigonometry is not limited to abstract calculations. It translates into real situations, such as engineering planning, navigation, distance calculation, and even modeling natural phenomena.

• Importance of Practice and Use of Formulas and Identities: Calculating sines, cosines, and tangents is the basis for solving more complex problems in trigonometry. Constant practice combined with knowledge of formulas and identities will allow students to enhance their skills in this discipline.

Exercises

1. Exercise 1: Given a right triangle with an angle of 45 degrees and a hypotenuse of 10 units, calculate the sine, cosine, and tangent of that angle.

2. Exercise 2: Using the fundamental identity of trigonometry, verify if it is true for the angle of 60 degrees. What do you conclude from this result?

3. Exercise 3: A building is 20 meters high. At what distance from the building, in a straight line, is the angle formed by the view of a person at ground level to the top of the building 30 degrees? (Hint: use the tangent)