LAW OF COSINES An adjustment to solve Oblique

LAW OF COSINES An adjustment to solve Oblique Triangles

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 with triangle into Imagine Triangle ABC, Quadrant I of a a, setb of coordinate axes opposite sides and c. such that A is at the Origin. Let point C be (u, v) Dropping C an altitude from C, the point where it intersects AB will be (u, 0). a b Now the orange Now consider the Forconsider the green triangle… Green triangle… A B (u, v) v u (0, 0) c-u (u, 0) (c, 0) It is a right triangle. c It is a right triangle. The height is v. The length is u. The length is c-u. From the orange triangle… DERIVING THE LAW OF COSINES

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. 1. SSS – When you are given all three side lengths of a triangle. 2. SAS – When you are given two sides and the included angle. THE LAW OF COSINES

1. Write down a list of all 6 parts of the triangle. 2. Fill in what you know. 3. Find the first missing piece by plugging the given information into the Law of Cosines. 4. Once you have found a complete “letter pair” you can switch to the Law of Sines* (which you will learn later) 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

In Triangle PMF, M=127 o, p=15. 78, and f=8. 54. Find the length of side m. First, make your answer space… Next, determine your situation (SAS) Next, Solve for your opposite side op= M= 34. 95 127 o F= 18. 05 of= P= 15. 78 m = 22. 00 8. 54 Finally, be sure to perform your “quick check” by comparing large-small angles to large-small sides. EXAMPLE 1

In Triangle XYZ, x=3, y=7, and z=9. Find the unknown measures of the triangle. First, make your answer space… Next, determine your situation (SSS) As long as you’ve found the largest Next, Solve for the LARGEST oppositeangle, angle. you can switch to Law of Sines. NEVER find the largest angle with inverse sine. Y= 16. 06 o 123. 20 o y= 3 7 Z= 40. 74 z= 9 X= o x= Finally, be sure to perform your “quick check” by comparing large-small angles to large-small sides. EXAMPLE 2

In Triangle ABC, a=5, b=8, and c=14. Find the unknown measures of the triangle. First, make your answer space… Next, determine your situation (SSS) Next, Solve for the LARGEST opposite angle. A= a= B= b= 5 8 C= c= 14 So, WHY did we get an error? Think back to geometry … Remember the triangle inequality… The two short sides must add up to be longer than the longest side, otherwise it cannot form a triangle. a + b = 13 and c = 14 EXAMPLE 3 So, NO TRIANGLE