What data determines a triangle?

Solving a triangle is getting its sides and angles from enough data. Always use the convention: $a,b,c$ opposite $A,B,C$. A non-degenerate triangle has positive sides and angles between 0 and $\pi$.

One side and two angles

Knowing $a,B,C$, calculate $A=\pi-B-C$. $B+C<\pi$ is required. The law of sines gives:

$$b=\frac{a\sin B}{\sin A},\qquad c=\frac{a\sin C}{\sin A}.$$

The data determines a triangle, minus position and reflection.

Two sides and the angle between them

Knowing $b,c,A$, with $0 $$a=\sqrt{b^2+c^2-2bc\cos A}.$$

Then find, for example, $B$ by the law of cosines and make $C=\pi-A-B$. Using arccosine for an angle avoids choosing between two angles of the same sine.

Three sides

First check $a $$A=\arccos\frac{b^2+c^2-a^2}{2bc},$$

with analogous formulas for B and C. For example, sides 3, 4 and 5 determine a right angle opposite the side 5.

Two sides and an angle not understood

Knowing $a,b,A$, the law of sines requires:

$$\sin B=q,\qquad q=\frac{b\sin A}{a}.$$

If $q>1$, there is no solution. If $q=1$, the only candidate is $B=\pi/2$. If $0 $$B_1=\arcsin q,\qquad B_2=\pi-B_1.$$

Each candidate is only accepted if $A+B_i<\pi$. For each accepted, $C_i=\pi-A-B_i$ and $c_i=a\sin C_i/\sin A$. The positive third angle filter is indispensable.

Complete geometric classification

For $0

Data Number of triangles
$a 0
$a=h$ 1, with $B=\pi/2$
$h 2
$a\ge b$ 1

If $\pi/2\le A<\pi$, there is a solution when $a>b$ and none when $a\le b$. The side opposite a right or obtuse angle must be the larger of the two given sides.

A case with two answers

For $A=30°$, $a=5$ and $b=8$, we have $h=4<5<8$. The equation gives $\sin B=4/5$. The candidates are approximately $53{,}13°$ and $126{,}87°$; both add up to $A$ less than $180°$.

The third angles are approximately $96{,}87°$ and $23{,}13°$. There are two distinct triangles that satisfy the data. A calculator returning only the main arcsine does not resolve the entire discussion on its own.

In everyday life

  • Width of a river. A base on one bank and two angles are enough to find the distance to the other bank.
  • Aviation. The speed in relation to the ground and the deviation of an airplane’s course in a crosswind come from the resolution of a triangle.
  • More than one answer. The ambiguous case appears in forensics and navigation: with two sides and an angle not understood, there may be two possible positions, and another measurement is needed to decide.
One side and two angles are enough: the width of a river without crossing it.

Exercises with resolution

1. If $A=30°$, $a=4$ and $b=10$, is there a triangle?

Check resolution 1

No. $\sin B=10\cdot(1/2)/4=5/4>1$ would be required.

2. If $A=120°$, $a=10$ and $b=6$, how many triangles are there?

Check resolution 2

One, since A is obtuse and $a>b$. By calculation, $q=3\sqrt3/10$; just $B_1=\arcsin q$ leaves $C>0$.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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