Chapter 6B: Scalene Triangles

Law of Sines, Law of Cosines, ambiguous cases, and non-right triangle problem setup.

Scalene Triangle Facts

  • A scalene triangle has three different side lengths and three different angle measures.
  • Use Law of Sines when you know a matching side-angle pair.
  • Use Law of Cosines for SAS side-finding or SSS angle-finding.
  • SSA can create an ambiguous case: zero, one, or two possible triangles.

Problems Covered

Law of SinesAAS, ASA, and some SSA triangles. Law of CosinesSAS side and SSS angle problems. Ambiguous SSARecognize the classical extraneous triangle case.

6B.1 Law of Sines

Introduce the formula first by matching each side with the angle across from it.

\dfrac{a}{\sin(A)}=\dfrac{b}{\sin(B)}=\dfrac{c}{\sin(C)}

Worked Example 6B.1.1 — Surveyor Triangle

In triangle ABC, $A=42^\circ$, $B=65^\circ$, and $a=18$. Find $b$.

  • Use the known matching pair $A$ and $a$.\dfrac{a}{\sin(A)}=\dfrac{b}{\sin(B)}
  • Substitute known values.\dfrac{18}{\sin(42^\circ)}=\dfrac{b}{\sin(65^\circ)}
  • Solve downward.b=\dfrac{18\bullet\sin(65^\circ)}{\sin(42^\circ)}b\approx24.4

6B.2 Law of Cosines

The Law of Cosines is the non-right triangle version of the Pythagorean theorem. The angle term corrects for the triangle not being right.

c^2=a^2+b^2-2ab\bullet\cos(C)
C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)

Worked Example 6B.2.1 — SAS Side Finder

Two sides of a triangle are 9 and 14, and the included angle is $50^\circ$. Find the opposite side.

  • Use side-finder form.c^2=a^2+b^2-2ab\bullet\cos(C)
  • Substitute values.c^2=9^2+14^2-2(9)(14)\bullet\cos(50^\circ)
  • Take the square root.c^2\approx113.0c\approx10.6

6B.3 Ambiguous SSA

SSA problems can produce an extra solution because sine has the same positive value in Quadrants I and II.

Solution Strategy

After finding an angle with inverse sine, test the supplement too. Keep it only if the triangle angle sum stays below $180^\circ$.