Using Sine and Cosine Rules to Find a Side or Angle in a Triangle

This page provides a decision-making guide for choosing between the Sine Rule and Cosine Rule when solving triangle problems. It helps students understand when to use sine and cosine rule based on the given information.

Highlight: The choice between Sine and Cosine Rules depends on what information is given and what needs to be found.

When finding a side:

• Use the Cosine Rule if you know two sides and the included angle.
• Use the Sine Rule if you know two angles and one side.

When finding an angle:

• Use the Cosine Rule if you know all three sides.
• Use the Sine Rule if you know two sides and an angle (not the included angle).

Definition: The Sine Rule is expressed as a/sin A = b/sin B = c/sin C, where lowercase letters represent sides and uppercase letters represent angles.

Definition: The Cosine Rule for finding a side is a² = b² + c² - 2bc cos A, and for finding an angle is cos A = (b² + c² - a²) / (2bc).

These formulas provide the foundation for solving various triangle problems, making them essential tools in trigonometry.

The Sine Rule

This page focuses on the application of the Sine Rule, which is used when we know two sides and one angle (not the included angle) or when we know two angles and one side of a triangle.

Sine and cosine rule formula: a/sin A = b/sin B = c/sin C

The page provides several examples of how to use sine rule to find angle and side lengths:

1. Finding the size of angle R in a triangle with sides 9cm and 3cm, and an angle of 75°.

   Example: Using the Sine Rule, we get: 9/sin 75° = 3/sin R sin R = (3 × sin 75°) / 9 R = sin⁻¹(0.429) ≈ 25.4°

2. Finding the length of side x in a triangle with a side of 4cm, and angles of 40° and 45°.

   Example: Applying the Sine Rule: x/sin 45° = 4/sin 40° x = (4 × sin 45°) / sin 40° x ≈ 4.46cm

The page also highlights an important property of the Sine Rule:

Highlight: When rearranging the Sine Rule, remember to "change side, change operation." This means if you move a term to the other side of the equation, you need to switch between multiplication and division.

These examples demonstrate the versatility of the Sine Rule in solving various triangle problems, making it a crucial tool for students studying trigonometry.