While talking with your little sister one day, the conversation turns to shapes. Your sister is only in junior high school, so while she knows some things about right triangles, such as the Pythagorean Theorem, she doesn't know anything about other types of triangles. You show her an example of an oblique triangle by drawing this on a piece of paper:

Fascinated, she tells you that she knows how to calculate the area of a triangle using the familiar formula

"Do you know how to find the lengths of sides of the triangle and the area?" she asks.

### Finding Solutions for Triangles

Finding the sides, angles, and area for right triangles is often learned in Algebra and/or Geometry. However, it is common to learn how to determine this information in non-right triangles in Trigonometry.

Below is a chart summarizing common triangle techniques. This chart describes the type of triangle (either right or oblique), the given information, the appropriate technique to use, and what we can find using each technique.

Type of Triangle: |
Given Information: |
Technique: |
What we can find: |
---|---|---|---|

Right | Two sides | Pythagorean Theorem | Third side |

Right | One angle and one side | Trigonometric ratios | Either of the other two sides |

Right | Two sides | Trigonometric ratios | Either of the other two angles |

Oblique | 2 angles and a non-included side (AAS) | Law of Sines | The other non-included side |

Oblique | 2 angles and the included side (ASA) | Law of Sines | Either of the non-included sides |

Oblique | 2 sides and the angle opposite one of those sides (SSA) – Ambiguous case | Law of Sines | The angle opposite the other side (can yield no, one, or two solutions) |

Oblique | 2 sides and the included angle (SAS) | Law of Cosines | The third side |

Oblique | 3 sides | Law of Cosines | Any of the three |

angles |

#### Solve for angles of the triangle

In

Since we are given all three sides in the triangle, we can use the Law of Cosines. Before we can solve the triangle, it is important to know what information we are missing. In this case, we do not know any of the angles, so we are solving for

Now, we will find

We can now quickly find

#### Solve for the lengths and angles of the triangle

In triangle

In this triangle, we have the SAS case because we know two sides and the included angle. This means that we can use the Law of Cosines to solve the triangle. In order to solve this triangle, we need to find side

Now that we know

To find

#### Solve for length and angles of the triangle

In triangle

This is an example of the ASA case, which means that we can use the Law of Sines to solve the triangle. In order to use the Law of Sines, we must first know

Now that we know

### Examples

#### Example 1

Earlier, you were asked how you might help your sister find the lengths of the sides and the area of a non-right triangle.

Since you know that two of the angles are

And repeating the process for the third side:

Now you know all three angles and all three sides. You can use Heron's formula or the alternative formula for the area of a triangle to find the area:

#### Example 2

Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have. If a triangle has no solution or two solutions, explain why.

AAS, Law of Sines, one solution

#### Example 3

Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have. If a triangle has no solution or two solutions, explain why.

SAS, Law of Cosines, one solution

#### Example 4

Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have. If a triangle has no solution or two solutions, explain why.

SSS, Law of Cosines, one solution

### Review

Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have.

a=3,b=4,C=71∘ , findc .a=8,b=7,c=9 , findA .A=135∘,B=12∘,c=100 , finda .- \begin{align*}a=12, b=10, A=80^\circ\end{align*}, find \begin{align*}c\end{align*}.
- \begin{align*}A=50^\circ, B=87^\circ, a=13\end{align*}, find \begin{align*}b\end{align*}.
- In \begin{align*}\triangle ABC\end{align*}, \begin{align*}a=15, b=19, c=20\end{align*}. Solve the triangle.
- In \begin{align*}\triangle DEF\end{align*}, \begin{align*}d=12, E=39^\circ, f=17\end{align*}. Solve the triangle.
- In \begin{align*}\triangle PQR\end{align*}, \begin{align*}P=115^\circ, Q=30^\circ, q=10\end{align*}. Solve the triangle.
- In \begin{align*}\triangle MNL\end{align*}, \begin{align*}m=5, n=9, L=20^\circ\end{align*}. Solve the triangle.
- In \begin{align*}\triangle SEV\end{align*}, \begin{align*}S=50^\circ, E=44^\circ, s=12\end{align*}. Solve the triangle.
- In \begin{align*}\triangle KTS\end{align*}, \begin{align*}k=6, t=15, S=68^\circ\end{align*}. Solve the triangle.
- In \begin{align*}\triangle WRS\end{align*}, \begin{align*}w=3, r=5, s=6\end{align*}. Solve the triangle.
- In \begin{align*}\triangle DLP\end{align*}, \begin{align*}D=52^\circ, L=110^\circ, p=8\end{align*}. Solve the triangle.
- In \begin{align*}\triangle XYZ\end{align*}, \begin{align*}x=10, y=12, z=9\end{align*}. Solve the triangle.
- In \begin{align*}\triangle AMF\end{align*}, \begin{align*}A=99^\circ, m=15, f=16\end{align*}. Solve the triangle.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.12.