Perbandingan Trigonometri Pada Segitiga Siku-siku - Matematika Wajib Kelas X

Introduction

This tutorial provides a comprehensive overview of trigonometric ratios in right triangles, specifically tailored for 10th-grade mathematics. By the end of this guide, you will understand the concepts of angles in both degrees and radians, learn how to convert between these two units, and explore the fundamental trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent.

Step 1: Understanding Angles

• Definition of Angles: An angle is formed by two rays originating from a common endpoint, known as the vertex.
• Types of Angles:
  • Acute (less than 90°)
  • Right (exactly 90°)
  • Obtuse (greater than 90° but less than 180°)

Step 2: Learning Degree Measurement

• Definition of Degrees: A degree is a unit of measurement for angles, where a full circle is 360°.
• Usage: Most everyday measurements of angles are expressed in degrees.

Step 3: Understanding Radian Measurement

• Definition of Radians: A radian is another unit for measuring angles, defined based on the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle.
• Conversion Formula:
  • To convert degrees to radians:

    Radians = Degrees × (π / 180)
    

  • To convert radians to degrees:

    Degrees = Radians × (180 / π)
    

Step 4: Converting Between Degrees and Radians

• Example of Converting Radians to Degrees:

  • Given an angle of π/3 radians:
    • Degrees = (π/3) × (180 / π) = 60°

• Example of Converting Degrees to Radians:

  • Given an angle of 45°:
    • Radians = 45 × (π / 180) = π/4

Step 5: Exploring Trigonometric Ratios

• Definition: Trigonometric ratios relate the angles of a triangle to the lengths of its sides.
• Primary Ratios:
  • Sine (sin): Opposite side / Hypotenuse
  • Cosine (cos): Adjacent side / Hypotenuse
  • Tangent (tan): Opposite side / Adjacent side
• Reciprocal Ratios:
  • Cosecant (csc): 1/sin
  • Secant (sec): 1/cos
  • Cotangent (cot): 1/tan

Step 6: Applying Trigonometric Ratios

• Example Problem: In a right triangle where one angle is 30° and the opposite side measures 5 units:
  • Calculate the hypotenuse using sine:

    sin(30°) = Opposite / Hypotenuse
    0.5 = 5 / Hypotenuse
    Hypotenuse = 10 units
    

Step 7: Practice Problems

• To solidify your understanding, try the following:
  • Calculate the sine, cosine, and tangent for angles of 30°, 45°, and 60°.
  • Convert the following:
    • 90° to radians
    • π/6 to degrees

Conclusion

By mastering the concepts of angles in both degrees and radians, and the basic trigonometric ratios, you can solve various problems related to right triangles. Practice the conversion formulas and trigonometric calculations to enhance your skills. For further study, explore additional problems and resources provided in the video description or join study groups for collaborative learning.