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

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.
 * Check that this follows from the law of sines.


 * 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
A triangle $$\Delta ABC$$ has:


 * Side $$\overline{BC}$$ of 30m
 * $$\angle ABC=30^\circ$$ and
 * $$\angle BCA=45^\circ$$

Draw a rough diagram.
 * What are the lengths of the remaining sides, and what is the missing angle?

A triangle $$\displaystyle \Delta ABC$$ has:


 * Side $$\overline{BC}$$ of 30m
 * $$\angle BCA=30^\circ$$ and
 * $$\angle CAB=45^\circ$$

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?

Matemática elementar/Trigonometria/Resolução de triângulos