Introduction

Relevance of the Topic

Trigonometry is one of the most important branches of mathematics. Its applications extend across various fields of knowledge, such as engineering, physics, architecture, and even visual arts. The study of Basic Trigonometric Lines (30º, 45º, 60º) provides the foundation to understand and solve problems involving triangles, angle formation, and distance and area calculations. These lines, also known as notable angles, are widely used in practical and theoretical contexts, becoming an essential tool in the kit of a well-prepared mathematician.

Contextualization

In the Mathematics curriculum of the 3rd year of High School, the study of Basic Trigonometric Lines is part of the Trigonometry unit, which in turn integrates the Geometry block. After learning fundamental trigonometric concepts, such as sine, cosine, and tangent, and their ratios in the right triangle, the focus now is to develop the ability to apply these concepts in concrete situations. The basic trigonometric lines constitute an essential part of this learning stage, serving as reference points that facilitate the resolution of problems involving triangles, angles, and distances. Understanding these lines and their applications prepares students to advance in the discipline, exploring more complex topics, such as identifying angles in any quadrant and interpreting trigonometric functions in the unit circle.

Theoretical Development

Components

• Trigonometric Line of 30º, 45º, and 60º: These are the three angles that make up the equilateral triangle. These angles are known as 'notable angles' and frequently appear in the resolution of various trigonometric problems. The direct consequence of them being in the equilateral triangle is that the sides opposite them have equal measures, facilitating the resolution of several problems.

• Trigonometric Expressions for 30º, 45º, and 60º: Due to the special characteristics of these angles, it is possible to determine their exact values for the trigonometric functions sine, cosine, and tangent. For 30º we have sin(30º) = 1/2, cos(30º) = √3/2, and tan(30º) = 1/√3. For 45º we have sin(45º) = √2/2, cos(45º) = √2/2, and tan(45º) = 1. Finally, for 60º we have sin(60º) = √3/2, cos(60º) = 1/2, and tan(60º) = √3.

Key Terms

• Notable Angles: These are the angles of 30, 45, and 60 degrees, considered notable for having special values for the trigonometric functions sine, cosine, and tangent.

• Trigonometric Line: This is the axis that connects the angle to the trigonometric circle, being used to determine the values of the trigonometric functions.

• Trigonometric Circle: Graphical representation used to visualize trigonometric relationships. Each point on this circle represents an angle, and the distance from this point to the center of the circle represents the angle measure.

Examples and Cases

• Case 1: If a right triangle has an angle of 30º, and the side opposite that angle measures 10 cm, we can use the tangent property to determine the measure of the other side. Knowing that tan(30º) = opposite side / adjacent side, we have tan(30º) = 10 / adjacent side. Using the trigonometric expression for 30º, we have 1/√3 = 10 / adjacent side. Isolating the adjacent side, we find the adjacent side = 10√3 / 3.

• Case 2: If in an equilateral triangle the side length is 4 cm, and we want to determine the height of this triangle, we can consider that the height is also the median, the bisector, and the median of the triangle. The height is the line that passes through the vertex and is perpendicular to the opposite side. In the equilateral triangle, the height is also the perpendicular bisector and the median. Therefore, the height is the side of the right triangle formed by the 30º angle. Thus, we can use the sine property to determine the height. We know that sin(30º) = opposite side / hypotenuse, and the hypotenuse is the side of the equilateral triangle, which is 4 cm. Therefore, the height of the equilateral triangle is sin(30º) * 4, which is 2 cm.

• Case 3: If in a right triangle the measure of the side adjacent to a 45º angle is 3 cm, we can use the sine expression to determine the measure of the hypotenuse. We know that sin(45º) = opposite side / hypotenuse. Since the triangle is right-angled, the remaining angle is 90º, and the sum of the internal angles of a triangle is always 180º, we can conclude that the angle opposite the hypotenuse is 45º, and therefore, the sine of this angle can be calculated. We have sin(45º) = opposite side (which is the hypotenuse) / hypotenuse. So, 1 = hypotenuse / hypotenuse. Therefore, the hypotenuse measures 3 cm. This is one of the main concepts that helps understand why in the isosceles triangle (one that has two equal sides) the angle opposite the equal side is always 45º.

These examples illustrate how basic trigonometric lines are fundamental for solving a variety of mathematical problems, making the learning of these concepts more tangible and applicable. With practice, the manipulation of these angles and their trigonometric functions will become increasingly natural and intuitive.

Detailed Summary

Key Points

• Importance of Notable Angles: The angles of 30º, 45º, and 60º, known as notable angles, are highly relevant in Trigonometry due to their special values for the sine, cosine, and tangent functions. These values, which are always decimals or radicals, are frequently used to solve real and theoretical problems.

• Applications in Triangles: Basic trigonometric lines have a direct application in solving problems with triangles, including determining the values of their sides and angles. For example, knowing that in the right triangle the sine is the ratio between the opposite side and the hypotenuse, and the cosine is the ratio between the adjacent side and the hypotenuse, we can use the values of the sine and cosine of 30º, 45º, and 60º to solve triangulation problems.

• Use in the Trigonometric Circle: The trigonometric circle is a tool that aids in the visualization and understanding of the values of trigonometric functions. In it, the notable angles are easily identified due to their positions and relationships with the x-axis and the unit radius.

Conclusions

• Foundation for Trigonometric Calculations: Understanding the basic trigonometric lines and how they relate to angles and sides of triangles provides the necessary foundation to perform trigonometric calculations.

• Ease of Problem Solving: Knowing how to correctly use the values of basic trigonometric lines simplifies the resolution of many problems and exercises involving Trigonometry.

• Practical Applications: The concepts and practices associated with these notable angles have a wide range of applications, from solving engineering problems to data analysis in social sciences.

Exercises

1. Given a right triangle with an angle of 60º, and the side opposite that angle measuring 8 cm, determine the measure of the adjacent side.
2. In an equilateral triangle with a side of 6 cm, calculate the height.
3. In an isosceles right triangle with a side of 2 cm, calculate the measure of the hypotenuse.