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.

The Cosine Rule

This page delves into the Cosine Rule, which is used in two specific scenarios: when given two sides and their included angle, or when all three sides of a triangle are known.

Cosine rule formula for side: a² = b² + c² - 2bc cos A

Cosine rule formula for angle: cos A = $b² + c² - a²$ / (2ac)

The page provides two key examples:

1. Calculating side length with cosine rule example: Find the length of side BC in a triangle where two sides are 7cm and 3cm, with an included angle of 35°.

   Example: Using the Cosine Rule: a² = 7² + 3² - 2(7)(3)cos(35°) a² = 49 + 9 - 34.40 a² = 23.60 a = √23.60 ≈ 4.9cm

2. Finding angle using Cosine Rule calculator: Find the size of angle B in a triangle with sides 4cm, 5cm, and 7cm.

   Example: Applying the Cosine Rule for angles: cos B = $4² + 5² - 7²$ / (2 × 4 × 5) cos B = $16 + 25 - 49$ / 40 cos B = -0.2 B = cos⁻¹$-0.2$ ≈ 101.5°

These examples illustrate how the Cosine Rule can be used to find both sides and angles in triangles, making it a versatile tool in trigonometry. The page emphasizes the importance of choosing the correct formula based on the given information and what needs to be calculated.

Highlight: When using the Cosine Rule, pay attention to whether you're solving for a side or an angle, as the formulas differ slightly.

Understanding when and how to apply the Cosine Rule is crucial for students tackling complex triangle problems in trigonometry and geometry.

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.