Understanding and Applying the Law of Sines

The law of sines, also known as the sine theorem, is a powerful tool in trigonometry for solving triangles. This page demonstrates its formula and practical applications through examples.

Definition: The law of sines states that for any triangle ABC with sides a, b, c and opposite angles A, B, C: a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle.

The formula is visually represented at the top of the page, showing its application to a triangle with sides a, b, c and corresponding angles A, B, C.

Example: The first example illustrates how to calculate the lengths of triangle sides and angles using the law of sines.

Given:

• One side length: 5 cm
• Two angles: 70° and 48°

The solution process involves:

1. Calculating the third angle (62°) using the fact that angles in a triangle sum to 180°.
2. Applying the law of sines to find the unknown sides.

Example: The second example demonstrates how to find the radius of the circumscribed circle of a triangle.

Given:

• One side length: 6 cm
• Two angles: 48° and 70°

The solution involves:

1. Calculating the third angle (62°).
2. Using the law of sines formula 2R = a/sin A to find the diameter and then the radius.

Highlight: The page concludes with the complete solution, stating that the triangle sides are approximately 5 cm, 5.9 cm, and 6.3 cm, with angles 62°, 48°, and 70°. The radius of the circumscribed circle is approximately 1.6 cm.

This comprehensive example showcases how the law of sines can be used to solve various aspects of triangle problems, including finding unknown sides, angles, and related circle properties. It's an essential tool for students learning trigonometry and geometry, particularly when dealing with non-right triangles.