Trigonometry/Solving Triangles

One of the most common applications of trigonometry is solving triangles—finding missing sides and/or angles given some information about a triangle. The process of solving triangles can be broken down into a number of cases.

Given a Right Triangle

If we know the triangle is a right triangle, one side and one of the non-right angles or two sides uniquely determine the triangle. (It is also worth noting that if we are given a triangle not specified to be a right triangle, but two given angles are complementary, then the third angle must be a right angle, so we have a right triangle.) Use right triangle trigonometry to find any missing information.

Given Three Angles (AAA) and No Sides

From two angles the third can be computed. Hence, given two angles one is given all three angles (since the sum of the measures of the internal angles of a triangle form a straight angle). Any two triangles with congruent corresponding angles are themselves congruent triangles. Congruent triangles are similar triangles, meaning that they possess precisely the same shape. However, without a length to extrapolate a relative size, no computation of sides, length or size can be performed.

Given Three Sides (SSS)

Given three sides of a triangle, we can use the Law of Cosines to find any of the missing angles (the alternate form $\cos B=\frac{a^2+c^2-b^2}{2ac}$ may be helpful here).

Given Two Sides and the Included Angle (SAS)

Given two sides and the angle included by the two given sides, we can apply the Law of Cosines to find the missing side, then procede as above to find the missing angles.

Given Two Angles and a Side (AAS or ASA)

Given two angles, we can find the third angle (since the sum of the measures of the three angles in a triangle is a straight angle). Knowing all three angles and one side, we can use the Law of Sines to find the missing sides.

Given Two Sides and an Angle Not Included by the Two Sides (SSA or the Ambiguous Case)

Here, we run into trouble—the given information may not uniquely determine a triangle. One of the two given sides is opposite the given angle, so we can apply the Law of Sines to try to find the angle opposite the other given side. When we do this (given sides $a$ and $b$ and angle $A$), we'll have $\sin B=\frac{b\sin A}{a}$ and there are three possibilities: $\sin B>1$, $\sin B=1$, or $\sin B<1$.

If $\sin B>1$, then there is no angle $B$ that meets the given information, so no triangle can be formed with the given sides and angle.

If $\sin B=1$, then we have a right triangle with right angle at $B$ and we can proceed as above for right triangles.

If $\sin B<1$, then there are two possible measures for angle $B$, one accute (the inverse sine of the value of $\sin B$) and one obtuse (the supplement of the accute one). The accute angle will give a triangle and we can find the missing information as above now that we have two angles. The obtuse angle may or may not give a triangle--when we attempt to compute the third angle, it may be that $A+B$ is more than a straight angle, leaving no room for angle $C$. If there is room for an angle $C$, then there is a second possible triangle determined by the given information and we can find the last missing length as above.