Using the Law of Sines and the Law of Cosines
to Solve Triangles

  1. If SSS
    1. Given sides a, b, and c,
      Use the Law of Cosines to determine mA.
    2. Use the Law of Cosines to determine mB.
    3. Use the sum of the angles of a triangle = 180 to find mC.

  2. If SAS
    1. Given sides a and b, and C,
      Use the Law of Cosines to determine side c.
    2. Use the Law of Cosines to determine B.
    3. Use the sum of the angles of a triangle = 180 to find mA.

  3. If ASA
    1. Given mA and mB and side c,
      Use the sum of the angles of a triangle = 180 to find mC.
    2. Use the Law of Sines to determine side b.
    3. Use the Law of Sines to determine side a.

  4. If AAS
    1. Given mA and mB and side a,
      Use the sum of the angles of a triangle = 180 to find mC.
    2. Use the Law of Sines to determine side b.
    3. Use the Law of Sines to determine side c.

  5. If SSA (Ambiguous Case)
    1. Given sides a and b, and A,
      Use the Law of Sines to solve for sin B.
      1. If sinB > 1,
        There is no triangle.
      2. If sinB 1,
        Determine mB in quadrant I.
        1. If mA + mB 180
          There is no triangle.
        2. If mA + mB < 180
          There is at least one triangle.
          1. Determine mB in quadrant II.
            It has the same sine value as B .
            Call this angle, B'.
          2. Determine mA + mB'
            1. If mA + mB' 180
              There is only one triangle.
              1. Determine mC using the sum of the angles in a triangle = 180
              2. Determine side c using the Law of Sines.
            2. If mA + mB' < 180
              There are two triangles.
              1. Determine mC using the sum of the angles in a triangle = 180
              2. Determine side c using the Law of Sines.
              3. Determine mC' using the sum of the angles in a triangle = 180
              4. Determine side c' using the Law of Sines.

The analysis of the Ambiguous Case was taken from a letter to The Mathematics Teacher by Carolyn J. Case, Vincennes University, Vincennes, IN.