Law of Sines Calculator

How to Calculate Using the Law of Sines

The Law of Sines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. It's particularly useful for solving problems involving non-right triangles when we know certain side lengths and angles.

Law of Sines Formula

For a triangle with sides a, b, and c, and angles A, B, and C opposite these sides respectively:

\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Where:

• a, b, c are the lengths of the sides of the triangle
• A, B, C are the angles opposite to sides a, b, c respectively
• sin A, sin B, sin C are the sines of angles A, B, C respectively

Calculation Steps

1. Identify the known values (at least three) among side lengths and angles.
2. Choose the appropriate form of the Law of Sines equation based on the known and unknown values.
3. Substitute the known values into the chosen equation.
4. Solve for the unknown side or angle.
5. If needed, use the results to calculate other unknown values.
6. Verify that the sum of all angles equals 180°.

Example Calculation

Let's solve a triangle with side a = 10 units, angle A = 30°, and side b = 15 units.

1. Use the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \)
2. Substitute the known values: \( \frac{10}{\sin 30°} = \frac{15}{\sin B} \)
3. Solve for angle B: \( \sin B = \frac{15 \sin 30°}{10} \)
4. Calculate: \( B = \arcsin(\frac{15 \times 0.5}{10}) \approx 48.59° \)
5. Find angle C: C = 180° - 30° - 48.59° = 101.41°
6. Calculate side c: \( c = \frac{10 \sin 101.41°}{\sin 30°} \approx 19.32 \) units

Visual Representation

This diagram illustrates the triangle from our example, showing how the Law of Sines relates the sides and angles.