The document discusses the Law of Cosines, which can be used to find the angles of a triangle when only the three sides are known. It provides the Law of Cosine formulas for finding the angles opposite sides a, b, and c. Examples are given for finding a missing side with two sides and the included angle (SAS) and for finding a missing angle with all three sides known (SSS). A word problem example finds the distance a golf ball is from the cup using the Law of Cosines.

1. Jim Smith
JCHS
Section 8-7

2. THE LAW OF COSINES ALLOWS US TO SOLVE THE TRIANGLE
WHEN WE CAN’T USE THE LAW OF SINES. SUPPOSE WE KNOW
THE 3 SIDES OF A TRIANGLE BUT NONE OF THE ANGLES.
THE LAW OF COSINES WILL HELP US FIND THE ANGLES.
a
b
c
A
C
B
a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos B
c² = a² + b² - 2ab cos C
NOTICE WHEN YOU START WITH SIDE a YOU END WITH ANGLE A
WHEN YOU START WITH SIDE b YOU END WITH ANGLE B
WHEN YOU START WITH SIDE c YOU END WITH ANGLE C

3. BE CAREFUL WHEN YOU PUT THIS IN YOUR CALCULATOR
a² = b² + c² - 2bc cos A
a = ( b² + c² - 2bc cos A

4. 2 SIDES AND THE INCLUDED ANGLE ( SAS )
A
B
c
6
8
a
FIND SIDE a
a² = b² + c² - 2bc cos A
a² = 8² + 6² - 2 ( 8 ) ( 6 ) cos 50
a = ( 8² + 6² - 2 ( 8 ) ( 6 ) cos 50
a = 6.2
50°

5. 3 SIDES ( SSS )
A
B
C
10
15
22
FIND ANGLE A
a² = b² + c² - 2bc cos A
15² = 22² + 10² - 2 ( 22 ) ( 10 ) COS A
225 = 484 + 100 – 440 COS A
225 = 584 – 440 COS A
-359 = - 440 COS A
-359 = COS A
-440
2nd COS ( -359 / -440
A = 35.3°

6. Golf On a 180-yard hole, a golfer hits the ball
175 yards, 3° off-line. How far is the ball from the cup?
A.10.6 yd C.42.1 yd
B.15.0 yd D.12.2 yd
180
175
3°
X
SAS
a² = b² + c² - 2 bc cos 3
a² = 180² + 175² - 2( 180 ) ( 175 ) cos 3
a = 180² + 175² - 2( 180 ) ( 175 ) cos 3
a = 10.55
from glencoe.com