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Section B.3 Oblique Triangle Trigonometry

An oblique triangle is any triangle which does not contain a right angle. As such, the rules of Right Triangle Trigonometry do not apply!

For an oblique triangle labeled as shown, the relations between the sides and angles are given by the Law of Sines and the Law of Cosines.

Subsection B.3.1 Law of Sines

\begin{equation} \frac{\sin a}{A} = \frac{\sin b}{B} =\frac{\sin c}{C} \tag{B.3.1} \end{equation}

The law of Sines is used when you know the length of one side, the angle opposite it, and one additional angle (SAA) or side (SSA). If this is not the case use the Law of Cosines.

Take care in the (SSA) situation. This is known as the ambiguous case, and you must be alert for it. It occurs because there are two angles between 0 and 180° with the same sine. When you use your calculator to find \(\sin^{-1}(x)\) it may return the supplement of the angle you want. In fact, there may be two possible solutions to the problem, or one or both solutions may be physically impossible and must be discarded.

If one of the angles is 90°, then the Law of Sines simplifies to the definitions of sine and cosine since the \(\sin(90°) = 1\text{.}\)

Subsection B.3.2 Law of Cosines

\begin{align} A^2 \amp= B^2 + C^2 - 2 B C \cos a\tag{B.3.2}\\ B^2 \amp= A^2 + C^2 - 2 A C \cos b\tag{B.3.3}\\ C^2 \amp= A^2 + B^2 - 2 A B \cos c\tag{B.3.4} \end{align}

Thea Law of Cosines is used when you know two sides and the included angle (SAS), or when you know all three sides but no angles (SSS). In any other situation, use the Law of Sines.

If one of the angles is \(\ang{90}\) the Law of Cosines simplifies to the Pythagorean Theorem since \(\cos (90°) = 0\text{.}\)