Introduction

Relevance of the Topic

Metric relations in the right triangle are one of the main tools in solving geometric problems and have numerous practical applications in trigonometry. These relations, based on the famous Pythagorean Theorem, allow calculating the length of sides and the measure of angles without the need for direct measuring tools, only with the knowledge about the relations between the sides in a right triangle.

Context

This topic falls within the scope of plane trigonometry, one of the main subdivisions of mathematics. In the curriculum, metric relations in the right triangle are addressed after understanding the Pythagorean theorem and before the more in-depth study of trigonometry and its applications. This is because understanding these relations is crucial for manipulating triangles in various areas of mathematics, physics, engineering, and even in everyday issues involving the understanding and use of measures and coordinates.

Theoretical Development

Components

• Right Triangle: The right triangle is a triangle that contains a 90-degree angle. The point where the perpendiculars meet is called the vertex. The metric relations we will address depend only on the lengths of the sides of this triangle and not on the angles.

• Hypotenuse: The hypotenuse is the longest side of a right triangle. It is opposite the right angle. In trigonometry, the hypotenuse is often referred to as side 'H'.

• Legs: The legs of a right triangle are the two sides that form the right angle. In trigonometry, the legs are often denoted as side 'a' and 'b'.

• Pythagorean Theorem: The Pythagorean theorem is a fundamental relation in the right triangle that states that the square of the hypotenuse is equal to the sum of the squares of the legs. Mathematically, this can be expressed as H² = a² + b², where H represents the hypotenuse and a and b represent the legs.

• Basic Metric Relations: From the Pythagorean theorem, a series of metric relations are derived that relate the lengths of the sides of the right triangle. These are the relations that will be studied, including the measurement of angles.

Key Terms

• Pythagorean Theorem: This is one of the most famous and used theorems in mathematics, especially in geometry. It provides a relationship between the lengths of the sides in a right triangle.

• Cosine: In the context of metric relations in the right triangle, the cosine of an angle is defined as the length of the leg adjacent to that angle divided by the length of the hypotenuse. Mathematically, this can be expressed as cos θ = a / H.

• Sine: The sine of an angle in the right triangle is the length of the leg opposite the angle divided by the length of the hypotenuse. Mathematically, this can be expressed as sin θ = b / H.

• Tangent: The tangent of an angle in the right triangle is the length of the leg opposite the angle divided by the length of the adjacent leg. Mathematically, this can be expressed as tan θ = b / a.

Examples and Cases

1. Example 1: Using the Pythagorean relation to find the hypotenuse

   • Given a right triangle with leg lengths a=3 and b=4, let's apply the Pythagorean theorem to find the measure of the hypotenuse, H.
   • By the formula of the Pythagorean theorem, H² = a² + b².
   • Substituting the values, we have H² = 3² + 4² = 9 + 16 = 25
   • Taking the square root on both sides, we get H = √25 = 5. Therefore, the hypotenuse measures 5 units in length.

2. Example 2: Using the cosine trigonometric function to find an angle

   • Assuming a right triangle with leg lengths a=4 and b=3, we want to find the cosine of the angle θ.
   • The cosine of an angle is given by cos θ = a / H, where H is the hypotenuse.
   • Applying the formula, we have cos θ = 4 / 5 = 0.8.
   • Therefore, the cosine of the angle θ is 0.8 or 80% in percentage notation.

3. Example 3: Using the tangent trigonometric function to find an angle

   • Considering a right triangle with leg lengths a=5 and b=12, let's find the tangent of the angle θ.
   • The tangent of an angle is given by tan θ = b / a.
   • Applying the formula, we have tan θ = 12 / 5 = 2.4.
   • Therefore, the tangent of the angle θ is 2.4.

Detailed Summary

Key Points

• Right Triangle and Pythagorean Theorem: Understanding that the right triangle is a special case where one of the angles measures 90 degrees (right angle), and the importance of the Pythagorean Theorem in this context. The theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse, that is, a² + b² = H².

• Hypotenuse and Legs: Knowing that the hypotenuse is the longest side of a right triangle and that the legs are the other two sides that form the right angle. In lecture notes, precise definitions are given for these terms, as well as visual examples to ensure understanding.

• Trigonometric Functions in the Right Triangle: Deepening the application of trigonometric functions (sine, cosine, and tangent) in right triangles. Each of these functions relates the measure of an angle of this triangle to the lengths of the sides.

• Relation between Trigonometric Functions and Angle Measurements: Understanding that the trigonometric functions cosine, sine, and tangent can be applied to the angles of a right triangle to find the ratio between the lengths of the sides. In other words, these functions directly link the concept of an angle with the concept of distance.

• Practical Applications: Metric relations in the right triangle are not just theory; they have practical applications in real-life problem-solving, especially in physics and engineering.

Conclusions

• Metric relations in the right triangle, based on the Pythagorean theorem, are fundamental concepts in mathematics. Understanding these relations is essential, as they are an essential step for the study of trigonometry and have practical applications in various areas.

• The ability to manipulate right triangles using metric relations gives students the ability to solve a variety of problems, from calculating inaccessible distances to sizing structures, among others.

• Mastering metric relations in the right triangle, along with practice in applying the concepts, will allow students to solve a wide variety of problems involving trigonometry.

Proposed Exercises

1. Exercise 1: Given a right triangle whose legs measure 7cm and 24cm, calculate the length of the hypotenuse using the Pythagorean theorem.

2. Exercise 2: In a right triangle, the cosine of the largest angle is equal to 0.6. If the leg opposite this angle measures 8 meters, what is the length of the hypotenuse?

3. Exercise 3: A right triangle has a hypotenuse of 10 meters and a leg of 6 meters. Calculate the sine of the angle opposite this leg.