Sine Rule Calculator for Triangle Problems

Choose what you want to find:

Find Missing Side

Known Side (a):

Opposite Angle to Known Side (A°):

Opposite Angle to Unknown Side (B°):

What is the Sine Rule?

The sine rule (also known as the law of sines) is a fundamental theorem in trigonometry that relates the sides of a triangle to the sines of its opposite angles. It states that in any triangle, the ratio of each side to the sine of its opposite angle is constant.

a/sin(A) = b/sin(B) = c/sin(C)

Where a, b, c are the sides and A, B, C are the opposite angles

When to Use the Sine Rule

The sine rule is particularly useful when you have:

Two angles and one side (AAS or ASA cases)
Two sides and a non-included angle (SSA case – may have ambiguous solutions)

Triangle Labelling Convention

In triangle ABC:

• Side ‘a’ is opposite angle A
• Side ‘b’ is opposite angle B
• Side ‘c’ is opposite angle C

Step-by-Step Method

1 Identify known values: Determine which sides and angles you know, and which you need to find.

2 Set up the sine rule: Write the formula using the known and unknown values.

3 Solve the equation: Cross-multiply and solve for the unknown value.

4 Check your answer: Verify that all angles sum to 180° and sides satisfy triangle inequality.

Worked Example

Example: Find the missing side

Given: In triangle ABC, angle A = 30°, angle B = 45°, side a = 10 cm

Find: Side b

Solution:

Using the sine rule: a/sin(A) = b/sin(B)

10/sin(30°) = b/sin(45°)

10/0.5 = b/0.707

20 = b/0.707

Therefore: b = 20 × 0.707 = 14.14 cm

The Ambiguous Case (SSA)

When given two sides and a non-included angle, there may be zero, one, or two possible triangles. This occurs when the given angle is acute and the side opposite to it is shorter than the other given side.

Common Applications

Navigation: Determining distances and bearings in triangulated positions
Surveying: Calculating land boundaries and property measurements
Engineering: Structural analysis and force calculations
Astronomy: Measuring celestial distances using parallax
Physics: Vector analysis and wave interference calculations

Frequently Asked Questions

Can I use the sine rule for right-angled triangles?

Yes, the sine rule works for all triangles, including right-angled ones. However, for right triangles, basic trigonometric ratios (SOH CAH TOA) are often simpler to use.

Why do I sometimes get two different answers?

This occurs in the ambiguous case (SSA) when two different triangles can satisfy the given conditions. Always check if both solutions are geometrically possible.

What units should I use for angles?

This calculator uses degrees. If you have angles in radians, multiply by 180/π to convert to degrees before inputting.

How accurate are the results?

Results are calculated to 4 decimal places for precision. For practical applications, round to an appropriate number of significant figures based on your input data accuracy.

Tips for Success

Always sketch the triangle and label known values clearly
Check that angles don’t exceed 180° and sides satisfy triangle inequality
Be aware of the ambiguous case when using SSA configuration
Use consistent units throughout your calculations
Verify your answer makes geometric sense