1. An adjustment to solve Oblique Triangles
LAW of CosINES
An adjustment to solve Oblique Triangles

2. An OBLIQUE triangle is a triangle that has NO right angle
An OBLIQUE triangle is a triangle that has NO right angle. Even though our trigonometric functions are based on ratios that exist in RIGHT TRIANGLES, we can still use them (with some adjustments) to solve a OBLIQUE TRIANGLES. By “Solving a triangle” we mean determining the measures of all six parts of the triangle; three angles and three sides.
OBLIQUE Triangles

3. Deriving the Law of Cosines
Given two sides of a triangle and their included angle (angle in between the two sides), or given all three sides of a triangle, we can use what is called the Law of Cosines to solve.
Now, superimpose that triangle into Quadrant I of a set of coordinate axes such that A is at the Origin.
Imagine Triangle ABC, with opposite sides a, b and c.
(u, v)
Let point C be (u, v) A C B c b a
Dropping an altitude from C, the point where it intersects AB will be (u, 0). v u
c-u
(0, 0)
(u, 0)
(c, 0)
Now consider the orange triangle…
For the green triangle…
Now consider the Green triangle…
It is a right triangle.
The height is v.
The length is u.
It is a right triangle.
The height is v.
The length is c-u.
From the orange triangle…
Deriving the Law of Cosines

4. There are two basic cases for using the Law of Cosines.
In either case, you will notice that you are given information for THREE different letters representing parts of the triangle.
SSS – When you are given all three side lengths of a triangle.
SAS – When you are given two sides and the included angle.
The Law of CoSines

5. Solving Using Law of CoSines
Write down a list of all 6 parts of the triangle.
Fill in what you know.
Find the first missing piece by plugging the given information into the Law of Cosines.
Once you have found a complete “letter pair” you can switch to the Law of Sines* if you wish.
*Note: If you begin using Law of Sines, make sure you have found the largest angle before you switch to the Law of Sines.
Solving Using Law of CoSines

6. In Triangle PMF, M=127o, p=15. 78, and f=8. 54
In Triangle PMF, M=127o, p=15.78, and f=8.54. Find the unknown measures of the triangle.
Example 1

7. In Triangle XYZ, x=3, y=7, and z=9
In Triangle XYZ, x=3, y=7, and z=9. Find the unknown measures of the triangle.
Example 2

8. In Triangle ABC, a=5, b=8, and c=14
In Triangle ABC, a=5, b=8, and c=14. Find the unknown measures of the triangle.
Example 3