Step By Step Calculus » 6.2 - General Triangles

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Synopsis
Sine Law: The Law of Sines states that if A,B,C A,B,C are the interior angles of a triangle and a,b,c a,b,c are the lengths of the sides opposite the angles A, B, C A, B, C respectively then
\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.
\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.
Ambiguity with Sine Law: The Sine Law can lead to ambiguous solutions meaning it may construct two different triangles based on the given information. It happens when the given information satisfies all the following conditions.
  • The only information known about the triangle is two sides and an angle opposite to one of the given sides.
  • The given angle is acute.
  • The side opposite to the given angle is shorter than at least one of the given sides.
Cosine Law: The Law of Cosines or Cosine Law states that if A,B,C A,B,C are the three interior angles of a triangle and a,b,c a,b,c are the lengths of the sides opposite the angles A,B,C A,B,C respectively, then
\displaystyle \displaystyle

\displaystyle a^2=b^2+c^2-2bc\cos A\displaystyle a^2=b^2+c^2-2bc\cos A
\displaystyle \displaystyle

\displaystyle b^2=a^2+c^2-2ac\cos B\displaystyle b^2=a^2+c^2-2ac\cos B
\displaystyle \displaystyle

\displaystyle c^2=a^2+b^2-2ab\cos C\displaystyle c^2=a^2+b^2-2ab\cos C
\displaystyle \displaystyle