Summary of Law of Sines for Oblique Triangles

Oblique Triangles

Oblique triangles are triangles that do not contain a right angle (90∘90^\circ). They can be categorized as either acute (all angles less than 90∘90^\circ) or obtuse (one angle greater than 90∘90^\circ). Standard trigonometric ratios for right triangles, such as SOH CAH TOA, cannot be applied directly to oblique triangles.

Triangle Notation

  • Angles: Represented by uppercase letters (AA, BB, and CC).

  • Sides: Represented by lowercase letters (aa, bb, and cc).

  • Opposition: In a standard triangle, side aa is opposite angle AA, side bb is opposite angle BB, and side cc is opposite angle CC.

Solving Oblique Triangles

Solving a triangle means finding the measures of all its sides and angles. This requires knowing at least one side length and any two other elements. There are four primary cases for oblique triangles:

  1. Law of Sines Cases:

    • Angle-Angle-Side (AAS): Two angles and a non-included side are known.

    • Angle-Side-Angle (ASA): Two angles and the side between them are known.

    • Side-Side-Angle (SSA): Two sides and an angle opposite one of them are known (the ambiguous case).

  2. Law of Cosines Cases:

    • Side-Angle-Side (SAS): Two sides and the included angle are known.

    • Side-Side-Side (SSS): All three sides are known.

Law of Sines

The Law of Sines states that the ratios of the lengths of sides to the sines of their opposite angles are equal:
asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Sum of Angles Identity:
The sum of the interior angles in any triangle is always 180∘180^\circ:
A+B+C=180∘A + B + C = 180^\circ

Example Problems

  1. Find a missing side (AAS):

    • Given: A=50∘A = 50^\circ, B=81∘B = 81^\circ, a=11a = 11 inches.

    • Procedure: Use the Law of Sines to find side bb.
      11sin⁡(50∘)=bsin⁡(81∘)\frac{11}{\sin(50^\circ)} = \frac{b}{\sin(81^\circ)}
      b=11⋅sin⁡(81∘)sin⁡(50∘)≈14.182 inchesb = \frac{11 \cdot \sin(81^\circ)}{\sin(50^\circ)} \approx 14.182 \text{ inches}

  2. Find a missing angle and side (ASA):

    • Given: A=49∘A = 49^\circ, B=61∘B = 61^\circ, a=12a = 12.

    • Step 1: Find angle CC using the angle sum property.
      C=180∘−(49∘+61∘)=70∘C = 180^\circ - (49^\circ + 61^\circ) = 70^\circ

    • Step 2: Use the Law of Sines to find side cc.
      12sin⁡(49∘)=csin⁡(70∘)\frac{12}{\sin(49^\circ)} = \frac{c}{\sin(70^\circ)}
      c=12⋅sin⁡(70∘)sin⁡(49∘)≈14.941c = \frac{12 \cdot \sin(70^\circ)}{\sin(49^\circ)} \approx 14.941