Understanding the sine and cosine rules is essential for solving problems involving non-right-angled triangles. These rules allow us to find unknown sides and angles in any triangle, expanding our problem-solving toolkit beyond the Pythagorean theorem, which applies only to right-angled triangles.

Introduction to the Sine Rule

The sine rule is a powerful tool for dealing with triangles where we don't have a right angle. It's especially useful when we have either two angles and one side (AAS or ASA) or two sides and a non-enclosed angle (SSA) information. The rule states:

where $a$, $b$, and $c$ are the lengths of the sides of the triangle, and $A$, $B$, and $C$ are the opposite angles.

Image courtesy of Cue Math

Example Using the Sine Rule

Given: $a = 7$ cm, $A = 35°$, and $B = 60°$, find $b$.

Solution:

1. Apply the sine rule:

2. Substitute the known values:

3. Solve the equation:

Therefore, the value of b is approximately $10.57$ cm.

Introduction to the Cosine Rule

The cosine rule helps us find a side of a triangle when we know the other two sides and the included angle, or an angle when we know all three sides. It's particularly useful for triangles that don't fit the criteria for using the sine rule. The cosine rule is expressed as:

This can also be rearranged to solve for an angle $A$:

Example Using the Cosine Rule

Given: $b = 8$ cm, $c = 6$ cm, and $A = 50°$, find $a$.

Solution:

1. Insert the known values into the cosine rule formula:

2. Solve for $a^2$:

3. Calculate $a$:

This process determines the length of side $a$ given the lengths of sides $b$ and $c$, and the angle $A$ between them.

Practice Problems

Problem 1: Using the Sine Rule to Find an Angle

Given a triangle with sides $a = 9$ cm, $b = 7$ cm, and $c = 5$ cm, and you know that angle $A = 70°$, find angle $B$.

Solution:

1. Use the sine rule:

2. Substitute known values:

3. Solve for $\sin B$ and then find $B$.

Problem 2: Using the Cosine Rule to Find a Side

You have a triangle where $b = 10$ cm, $c = 7$ cm, and the angle between them $A = 65°$. Find the length of side $a$.

Solution:

1. Apply the cosine rule:

2. Calculate $a^2$ and then $a$.

Tips for Using the Sine and Cosine Rules

• Identify whether to use the sine or cosine rule based on the given information.
• Accuracy is key when using your calculator, ensure it's set to the correct mode (degrees or radians) based on the angle measurements you're working with.
• Practice with different triangle configurations to become more comfortable with these rules.