Sine and Cosine Rules in Trigonometry

This page provides a comprehensive overview of the sine and cosine rules, essential tools in trigonometry for solving non-right-angled triangles. The document begins by presenting the formulas for both rules and then demonstrates their application through a detailed example.

Definition: The sine rule is a formula used in trigonometry to find unknown sides or angles in non-right-angled triangles.

The sine rule is presented in two forms:

1. For finding sides: a/sin A = b/sin B = c/sin C
2. For finding angles: sin A/a = sin B/b = sin C/c

Definition: The cosine rule is a formula used in trigonometry to find the length of a side or the size of an angle in a triangle when two sides and the included angle are known.

The cosine rule is presented in three equivalent forms:

1. a² = b² + c² - 2bc cos A
2. b² = a² + c² - 2ac cos B
3. c² = a² + b² - 2ab cos C

Example: The document provides a detailed example of how to apply cosine rule in trigonometry step by step. The problem involves a triangle with sides of 28cm and 22cm, and an included angle of 97°. The task is to find the length of the third side.

The solution process is broken down into four steps:

1. Label the sides of the triangle.
2. Write the appropriate cosine rule formula.
3. Substitute the known values into the formula.
4. Solve the equation to find the unknown side length.

Highlight: The example demonstrates the practical application of the cosine rule, showing how it can be used to solve complex triangular problems when traditional methods like the Pythagorean theorem are not applicable.

This comprehensive guide serves as an excellent resource for students learning when to use sine rule and cosine rule in various trigonometric problems. It provides a simple explanation of sine rule with examples and illustrates how to apply cosine rule in trigonometry step by step, making it valuable for both classroom instruction and self-study.