Understanding the Sine Rule for Non-Right-Angled Triangles

The sine rule is a powerful tool in trigonometry for solving problems involving non-right-angled triangles. This page demonstrates how to use the sine rule for finding missing sides and angles in various scenarios.

Definition: The sine rule states that for any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles.

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.

Finding Missing Sides

The page provides an example of using the sine rule in non-right angled triangles to find a missing side:

Example: In a triangle with side lengths 5cm and 4cm, and angles 35° and 32°, we can find the length of the third side (y) using the sine rule.

The solution process is as follows:

1. Set up the sine rule equation: y / sin(35°) = 4 / sin(32°)
2. Cross-multiply: y * sin(32°) = 4 * sin(35°)
3. Solve for y: y = (4 * sin(35°)) / sin(32°)
4. Calculate: y ≈ 4.348cm

Highlight: When using a sine rule calculator for finding missing sides, always ensure you're using the correct angles and sides in your equation.

Finding Missing Angles

The page also demonstrates how to use the sine rule to find missing angles:

Example: In a triangle with sides 5cm and 4cm, and one known angle of 20°, we can find another angle (B) using the sine rule.

The solution process is:

1. Set up the sine rule equation: 5 / sin(20°) = 4 / sin(B)
2. Cross-multiply: 5 * sin(B) = 4 * sin(20°)
3. Solve for sin(B): sin(B) = (4 * sin(20°)) / 5
4. Use inverse sine (arcsin) to find B: B = arcsin((4 * sin(20°)) / 5)
5. Calculate: B ≈ 15.95°

Vocabulary: The inverse sine function, also known as arcsin or sin^(-1), is used to find an angle when given its sine value.

This page provides a comprehensive overview of how to apply the sine rule for finding missing sides and angles in non-right-angled triangles, making it an invaluable resource for students studying trigonometry.