Key Concepts & Glossary

Key Equations

Law of Sines	[latex]\begin{array}{l}\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}\hfill \\ \frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }\hfill \end{array}[/latex]
Area for oblique triangles	[latex]\begin{array}{r}\hfill \text{Area}=\frac{1}{2}bc\sin \alpha \\ \hfill \text{ }=\frac{1}{2}ac\sin \beta \\ \hfill \text{ }=\frac{1}{2}ab\sin \gamma \end{array}[/latex]

The Law of Sines can be used to solve oblique triangles, which are non-right triangles.
According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution.
The ambiguous case arises when an oblique triangle can have different outcomes.
There are three possible cases that arise from SSA arrangement—a single solution, two possible solutions, and no solution.
The Law of Sines can be used to solve triangles with given criteria.
The general area formula for triangles translates to oblique triangles by first finding the appropriate height value.
There are many trigonometric applications. They can often be solved by first drawing a diagram of the given information and then using the appropriate equation.

altitude: a perpendicular line from one vertex of a triangle to the opposite side, or in the case of an obtuse triangle, to the line containing the opposite side, forming two right triangles

ambiguous case: a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle

Law of Sines: states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is equal to the remaining two ratios of angle measure to opposite side; any pair of proportions may be used to solve for a missing angle or side