Law of Sines Calculator (Sin Triangle)

Solve triangles using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).

Enter Known Values

Enter at least one side-angle pair plus one more known value. Leave unknowns blank.

Side a & Angle A (opposite)
Side b & Angle B (opposite)
Side c & Angle C (opposite)

Result

Triangle Solution
--
a/sin(A) = b/sin(B) = c/sin(C)
Side a--
Side b--
Side c--
Angle A--
Angle B--
Angle C--
Ratio (a/sinA)--
Area--

Step-by-Step Solution

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

Understanding the Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This relationship is written as a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides and A, B, C are the opposite angles respectively.

When to Use the Law of Sines

AAS (Angle-Angle-Side)

Two angles and a non-included side are known. Find the remaining side and angle.

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

ASA (Angle-Side-Angle)

Two angles and the included side are known. The third angle is 180 - A - B.

C = 180 - A - B

SSA (Side-Side-Angle)

Two sides and an angle opposite one of them. May have 0, 1, or 2 solutions (ambiguous case).

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

Triangle Area

Once all sides and angles are known, the area can be computed using the sine area formula.

Area = (1/2) a b sin(C)

The Ambiguous Case (SSA)

When given two sides and an angle opposite one of them, there may be zero, one, or two valid triangles. This is known as the ambiguous case. If the computed sine of the unknown angle exceeds 1, no triangle exists. If it equals 1, exactly one right triangle exists. Otherwise, two possible angles (supplementary) may both yield valid triangles.

Key Facts

  • The sum of all angles in a triangle is always 180 degrees.
  • The Law of Sines works for all triangles, not just right triangles.
  • For right triangles, the Law of Sines simplifies since sin(90) = 1.
  • The circumradius R of the triangle equals a / (2 sin(A)).