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 ASA (Side and angles at each end of that side)

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

The missing angle, $\theta$ is given by:

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

• 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 $c$ , then the side opposite the angle $\alpha$ has length $a$ given by:

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

• Check that this follows from the law of sines.

Now is also a good time to check that you can derive the law of sines for yourself. If given a problem triangle like this one in an exam, you might be asked both to find its missing sides and to derive the law of sines. Template:Robox/Close

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

Given AAS (one side and two other angles)

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

Exercises

Template:ExerciseRobox A triangle $\mathrm{\Delta }ABC$ has:

Side $\overline{BC}$ of 30m

$\mathrm{\angle }ABC=30^{\circ }$ and

$\mathrm{\angle }BCA=45^{\circ }$

Draw a rough diagram.

• What are the lengths of the remaining sides, and what is the missing angle?

Template:ExerciseRobox A triangle $\mathrm{\Delta }ABC$ has:

Side $\overline{BC}$ of 30m

$\mathrm{\angle }BCA=30^{\circ }$ and

$\mathrm{\angle }CAB=45^{\circ }$

Draw a rough diagram.

• What are the lengths of the remaining sides, and what is the missing angle?

Are these two triangles in Exercise 1 and Exercise 2:

• Similar to each other?
• Congruent to each other?

Template:Trig/NAV