Trigonometry/Solving Triangles Given ASA

One of the most common applications of trigonometry is solving triangles—finding missing sides and/or angles given some information about a triangle. The area of the triangle may also be required.

We can reconstruct the triangle given:

• ASA Angle-Side-Angle (see illustration below).
• AAS Angle-Angle-Side (see later on this page).
• SAS Side-Angle-Side (see next page).
• SSA Side-Side-Angle (later in this book). In the SSA case we may have one, two or no possible solutions.

• Given two angles, we can find the third angle (since the sum of the measures of the three angles in a triangle is ${\displaystyle 180^{\circ }}$ .

The missing angle, ${\displaystyle \theta }$ is given by:

${\displaystyle \theta =180^{\circ }-(\alpha +\beta )}$

• Knowing all three angles and one side, we can use the Law of Sines to find the missing sides.

In the illustration if the side we are given is the base and has length ${\displaystyle c}$ , then the side opposite the angle ${\displaystyle \alpha }$ has length ${\displaystyle a}$ given by:

${\displaystyle a=c\cdot {\frac {\sin(\alpha )}{\sin(\theta )}}}$

• The area may then be found from Heron's Formula, or more easily by the formulae given in Law of Sines.

Again we can work out the missing angle, since they sum to ${\displaystyle 180^{\circ }}$ . From there on we have the same information as after the first step in the ASA case.

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