Solving Triangles

Once you understand the Law of Sines and the Law of Cosines, you can use these formulas to solve any triangle that has at least three pieces of information.

To solve a triangle means that you find the measure of each angle and the length of each side.

Depending on what information you are given, you must start with either the Law of Sines or the Law of Cosines.

If you know..... Use.....

SSS Law of Cosines

SAS Law of Cosines

SSA Law of Sines*

AAS Law of Sines

ASA Law of Sines

*Watch for the ambiguous case

Solve this triangle:

$Solving triangles img 1$

We need to find $Solving triangles img 10$ , $Solving triangles img 2$ , and the length of side z. Let's take them one at a time.

1) First we'll use the Law of Cosines to find the length of side z.

We know we need to use the Law of Cosines because we are have SAS (two sides and the angle in between.)

C will be $Solving triangles img 3$ or 45°. a and b are 9 and 8. Let's substitute and solve:

$Solving triangles img 4$

$Solving triangles img 5$

$Solving triangles img 6$

$Solving triangles img 7$

$Solving triangles img 8$

$Solving triangles img 9$

2) Now we need to find $Solving triangles img 10$

Let's input our new info into our triangle:

$Solving triangles img 11$

We can use either the Law of Sines or the Law of Cosines. Let's use the Law of Sines (rewritten to portray angles X, Y, and Z instead of A, B, and C)

$Solving triangles img 12$

$Solving triangles img 13$

$Solving triangles img 14$

8 $Solving triangles img 15$

$Solving triangles img 16$

$Solving triangles img 17$

$Solving triangles img 18$

3) Find $Solving triangles img 19$

Since we know that the three angles of a triangle add up to 180°, we can subtract the two known angles to find the third:

180° - 45° - 59° = 76°

$Solving triangles img 20$

Now this answer assumes we found $Solving triangles img 21$ correctly. To double check your answer, you can find $Solving triangles img 2$ using either the Law of Sines or the Law of Cosines. If that answer also comes out around 76°, you can be fairly confident that you did everything correctly. Note, however, that both answers are dependent upon your calculation of side z, and z was rounded to the tenths place. Thus, there will be some variety in the answer due to rounding error. To reduce this, keep the value for z in the memory of your calculator and use this value instead of the rounded 6.6 in the final calculation.

Practice: Solve these triangles. Round all answers to the nearest tenth.

#1-3: Solve triangle XYZ

#4-6: Solve triangle DEF

$Solving triangles img 22$

Answers: (Note: Answers may vary slightly due to differences in rounding.)

#1-3) $Solving triangles img 23$ $Solving triangles img 24$ #4-6) $Solving triangles img 25$ f = 5.8 cm.

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