TFT

Law of Sines Calculator – Solve Triangles Using Sine Rule

Solve any triangle using the Law of Sines with our free online calculator. Find missing sides and angles for ASA, AAS, and SSA triangle configurations with step-by-step solutions.

Examples:

Understanding the Law of Sines

The Law of Sines relates the sides of a triangle to the sines of their opposite angles. It works for any triangle, not just right triangles. This makes it invaluable for solving triangles where you don't have a right angle to work with.

Use the Law of Sines when you know either two angles and one side (ASA or AAS), or two sides and a non-included angle (SSA). For SSA, there might be zero, one, or two possible triangles – this is called the "ambiguous case."

The Law of Sines Formula

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

Each side divided by the sine of its opposite angle equals the same value (which equals 2R, where R is the circumradius of the triangle).

To find a side:

a = b × sin(A) / sin(B)

To find an angle:

sin(A) = a × sin(B) / b

Worked Examples

Example 1: ASA Triangle

Given: ∠A = 45°, ∠B = 60°, a = 10
Find ∠C: C = 180° - 45° - 60° = 75°
Find b: b = 10 × sin(60°) / sin(45°)
b = 10 × 0.866 / 0.707 ≈ 12.25
Find c: c = 10 × sin(75°) / sin(45°)
c = 10 × 0.966 / 0.707 ≈ 13.66

Example 2: AAS Triangle

Given: ∠A = 30°, ∠B = 100°, a = 8
Find ∠C: C = 180° - 30° - 100° = 50°
Find b: b = 8 × sin(100°) / sin(30°)
b = 8 × 0.985 / 0.5 ≈ 15.76
Find c: c = 8 × sin(50°) / sin(30°)
c = 8 × 0.766 / 0.5 ≈ 12.26

Example 3: Equilateral Triangle Check

Given: ∠A = 60°, ∠B = 60°, a = 5
Find ∠C: C = 180° - 60° - 60° = 60°
All angles are 60°, so it's equilateral!
b = 5 × sin(60°) / sin(60°) = 5
c = 5 × sin(60°) / sin(60°) = 5
All sides equal 5, confirming equilateral triangle.

Quick Fact

The Law of Sines was known to Persian mathematicians in the 10th century. Al-Jayyani wrote the first general treatment of spherical trigonometry (which includes the Law of Sines) in his 11th-century book "The Book of Unknown Arcs of a Sphere."

Frequently Asked Questions

What's the ambiguous case (SSA)?

When you know two sides and a non-included angle, there might be 0, 1, or 2 possible triangles. This happens because sin(θ) = sin(180°-θ), so an angle and its supplement have the same sine value.

When should I use Law of Sines vs Law of Cosines?

Use Law of Sines for ASA, AAS, or SSA. Use Law of Cosines for SSS or SAS. Law of Sines is computationally simpler when it applies.

Can I use Law of Sines for right triangles?

Yes! For a right triangle with C = 90°, sin(C) = 1, so c = a/sin(A) = b/sin(B). This gives the familiar relationships from right triangle trigonometry.

What if I get sin(A) > 1?

That's impossible – sine values are always between -1 and 1. If your calculation gives sin(A) > 1, no triangle exists with the given measurements. Check your inputs.

How do I find the area using Law of Sines?

Area = ½ab×sin(C) = ½bc×sin(A) = ½ac×sin(B). Pick the formula using the two sides and included angle you know or have calculated.

Other Free Tools

Law of Cosines Calculator – Solve Triangles Using Cosine Rule

Solve triangles using the Law of Cosines with our free online calculator. Find missing sides and angles for SSS and SAS configurations with detailed step-by-step solutions.

Triangle Solver – Solve Any Triangle SSS SAS ASA AAS

Solve any triangle completely using SSS, SAS, ASA, or AAS methods with our free online triangle solver. Find all missing sides, angles, and area with step-by-step trigonometric solutions.

Triangle Angle Calculator – Find Missing Angles in a Triangle

Find any missing angle in a triangle with our free online angle calculator. Enter known angles or sides and instantly solve for the remaining angles using trigonometry rules.

Trig Function Calculator – Calculate Sin Cos Tan Online

Calculate any trigonometric function value including sin, cos, tan, csc, sec, and cot for any angle in degrees or radians with our free online trig calculator.

Inverse Trig Calculator – Find arcsin arccos arctan Online

Calculate inverse trigonometric functions including arcsin, arccos, and arctan with our free online inverse trig calculator. Get angle results in both degrees and radians.

Trig Identity Verifier – Verify Trigonometric Identities Online

Verify any trigonometric identity online with our free trig identity verifier. Simplifies both sides of an equation to check if the identity holds using fundamental trig rules.

Right Triangle Calculator – Solve Right Triangles Online

Solve any right triangle by entering two known values with our free online right triangle calculator. Find all sides, angles, area, and perimeter with clear step-by-step solutions.

Angle of Elevation & Depression Calculator – Solve Word Problems

Calculate the angle of elevation or depression with our free online solver. Enter height and distance to find the angle, or the angle to find missing dimensions – perfect for trig word problems.