Law of Sines Calculator Formula

Understand the math behind the law of sines calculator. Each variable explained with a worked example.

Use the Law of Sines Calculator

Formulas Used

Side B

side_b = side_a * sin(angle_b * pi / 180) / sin(angle_a * pi / 180)

Angle C

angle_c = 180 - angle_a - angle_b

Side C

side_c = side_a * sin((180 - angle_a - angle_b) * pi / 180) / sin(angle_a * pi / 180)

Variables

VariableDescriptionDefault
side_aSide a10
angle_aAngle A (degrees)(deg)30
angle_bAngle B (degrees)(deg)60

How It Works

Law of Sines

Formula

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

This law relates the sides of any triangle to the sines of their opposite angles.

Solving for Side b

b = a × sin(B) / sin(A)

When to Use

Use the Law of Sines when you know:

  • Two angles and one side (AAS or ASA)
  • Two sides and an angle opposite one of them (SSA, but watch for the ambiguous case)

    • Worked Example

    Find side b: a = 10, angle A = 30°, angle B = 60°.

    side_a = 10angle_a = 30angle_b = 60
    1. 01b = 10 × sin(60°) / sin(30°)
    2. 02= 10 × 0.8660 / 0.5
    3. 03= 17.3205
    4. 04Angle C = 180 - 30 - 60 = 90°

    Frequently Asked Questions

    What is the law of sines?

    The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C).

    What is the ambiguous case?

    When given two sides and a non-included angle (SSA), there may be zero, one, or two possible triangles. This is the ambiguous case of the law of sines.

    When should I use law of sines vs law of cosines?

    Use law of sines for AAS, ASA, or SSA problems. Use law of cosines for SAS or SSS problems.

    Learn More

    Guide

    Trigonometry Basics - Complete Guide

    Learn the fundamentals of trigonometry including sine, cosine, tangent, the unit circle, identities, and the laws of sines and cosines with practical examples.

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