Aturan Sinus dan Aturan Cosinus || Matematika Wajib kelas 10

Introduction

This tutorial focuses on the sine and cosine rules in trigonometry, essential concepts for 10th-grade mathematics. Understanding these rules is crucial for solving various problems related to triangles, particularly non-right triangles. This guide will break down the sine and cosine rules, providing clear explanations and practical examples.

Step 1: Understanding the Sine Rule

The sine rule relates the lengths of the sides of a triangle to the sines of its angles.

Formula

The sine rule is expressed as:

a / sin(A) = b / sin(B) = c / sin(C)

Where:

• a, b, and c are the lengths of the sides opposite to angles A, B, and C respectively.

Application

• Use the sine rule when you have:
  • Two angles and one side (AAS or ASA).
  • Two sides and a non-included angle (SSA).

Example Problem

1. Given a triangle with angles A = 30° and B = 45°, and side a = 10.
2. Find side b:
   • Calculate angle C: C = 180° - (A + B) = 180° - (30° + 45°) = 105°.
   • Apply the sine rule:

     10 / sin(30°) = b / sin(45°)
     b = (10 * sin(45°)) / sin(30°)
     

Practical Tip

• Make sure your calculator is set to the correct mode (degrees or radians) corresponding to your angle measurements.

Step 2: Understanding the Cosine Rule

The cosine rule is used to find a side or angle in a triangle.

Formula

The cosine rule is expressed as:

c² = a² + b² - 2ab * cos(C)

Where:

• c is the side opposite angle C, and a and b are the other two sides.

Application

• Use the cosine rule when you have:
  • All three sides (SSS).
  • Two sides and the included angle (SAS).

Example Problem

1. Given a triangle with sides a = 7, b = 10, and angle C = 60°.
2. Find side c:
   • Apply the cosine rule:

     c² = 7² + 10² - 2 * 7 * 10 * cos(60°)
     

Common Pitfalls

• Ensure that the angle C is measured in the correct unit.
• Be careful with negative values; cosine can yield negative results depending on the angle.

Step 3: Practice Exercises

To reinforce your understanding, practice with the following exercises:

1. Given angles A = 50° and B = 70°, and side a = 12, find side b using the sine rule.
2. In a triangle where a = 8, b = 6, and C = 45°, use the cosine rule to find side c.

Conclusion

Understanding the sine and cosine rules is fundamental in solving triangle-related problems in geometry. By applying these rules, you can determine unknown sides and angles in triangles. Practice consistently to improve your skills, and refer back to this guide as needed. For further practice, explore additional resources and exercises to strengthen your grasp of these concepts.