Summary of Trigonometric Ratios

Class Notes: Trigonometric Ratios - When the Math Circuit Makes a Full Turn!

Prof. Dr. Matias, the guide of the trigonometric circle!

Introduction

Greetings, young trigonometric enthusiasts, and welcome to another exciting journey through the world of Mathematics! Today, our focal point is Trigonometric Ratios, one of the cornerstones not only of Trigonometry but of the entire mathematical universe. Let's embark on a journey that unfolds around the circle - a circle that is always at the center of all trigonometric things. So, hold onto your compasses, this is a module you don't want to miss!

Relevance of the Topic

Trigonometric ratios are a powerful set of tools that allow us to explore the relationships between angles and the sides of a triangle. This topic is crucial for understanding and solving many practical and theoretical problems in mathematics and other disciplines, such as physics, engineering, computer science, and statistics. Trigonometric ratios are also the basis for many other topics in trigonometry, such as identities, equations, and trigonometric functions.

Theoretical Development

• Positioning in the Circuit: First, we need to position ourselves on the circle. In Trigonometry, our units of measurement are angles, and to map these angles, we use the trigonometric circle, a circle of unit radius (1 unit). The center of this circle is the origin O, and at this point, we have a landmark, the point (1,0), which indicates the beginning of our journey around the circle.

• We Love Angles!: An angle is a measure of the opening between two lines in the same region of the plane. Here, however, we need to deal with them only in the context of the trigonometric circle. We always measure angles counterclockwise from the positive x-axis. Angles can range from 0° to 360° or from 0 to 2π radians, a full trip around our circle.

• Intensive Friends: Angles and Sides in a Triangle!: Now we invite triangles to our trigonometric fireworks! In any triangle, there is always an angle with respect to which we are measuring the trigonometric ratios. The other two angles are called complementary angles to this angle. The basis of trigonometric ratios are the sides of the triangle that are relative to the angle in question.

• Trigonometric Ratios in Action!: Finally, we enter the core of the topic. The three main trigonometric ratios are called sine, cosine, and tangent, and are respectively expressed as the ratio between the lengths of the sides of the triangle relative to the angle under consideration and the hypotenuse, or the adjacent side, or the side opposite the angle in question.

  • Sine (sin): It is the ratio between the side opposite the angle and the hypotenuse, that is, sin(θ) = O/H.

  • Cosine (cos): Represents the ratio between the side adjacent to the angle and the hypotenuse, which mathematically is written as cos(θ) = A/H.

  • Tangent (tan): It is the ratio between the side opposite and the side adjacent to the angle, that is, tan(θ) = O/A.

  Let's practice! How about a challenge where we need to use these ratios to solve a real-world problem?

Detailed Summary

• Key Points:

  • Trigonometric Circle and Angles: The trigonometric circle is an essential tool in angle measurement. In it, we measure angles counterclockwise from the positive x-axis to the line representing the angle. The complete circle represents 360° or 2π radians, and this full rotation is what we call a full turn.

  • Trigonometric Ratios and Triangles: Trigonometric ratios are relationships between the sides of a right triangle and are used to calculate unknown angles or sides.

  • Sine (sin), Cosine (cos), and Tangent (tan): These are the three main trigonometric ratios. Sin (θ) is the ratio between the opposite side and the hypotenuse. Cos (θ) is the ratio between the adjacent side and the hypotenuse. Tan (θ) is the ratio between the opposite side and the adjacent side.

• Conclusions:

  • Universal Utility of Trigonometric Ratios: Trigonometric ratios play a fundamental role in many areas of mathematics and physics, allowing us to determine lengths and angles that would otherwise be inaccessible.

• Proposed Exercises:

  1. Find the value of sin(30°), cos(45°), and tan(60°) using the trigonometric circle and the triangle.

  2. In a right triangle, one of the angles is 30° and the side opposite this angle has a length of 1.5. What is the length of the hypotenuse and the adjacent side to this angle? Use trigonometric ratios to solve this problem.

  3. The side opposite a 60° angle in a right triangle has a length of 2. If the angle is the only known, what are the trigonometric ratios of this angle?

Oh, trigonometry... a mathematical siren's song that takes you on an epic journey around the circle, while tracing the hidden relationships between angles and sides of triangles. Remember that, despite all its complexities, trigonometry has deep and practical applications in every corner of reality. Embrace the circle, unravel the triangles, and let the trigonometric ratios guide you!