29
The law of cosines
Textbook: pp. 86–87
GoalKnow the law of cosines, use it to find the third side and the cosine of an angle, and see that it generalises the Pythagorean theorem.
New words
law of cosines · kosinuslar teoremasigeneralisation · umumlashmasquare of a side · tomon kvadraticosine of the angle · burchak kosinusi
Explanation
The law of cosines: the square of any side of a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of the angle between them: a² = b² + c² − 2bc·cosA. The proof drops a height and uses the Pythagorean theorem twice. If ∠A = 90°, then cos90° = 0 and the law becomes the Pythagorean theorem. For an obtuse angle cosA is negative, so a² is larger. When the sides are known, an angle is found from cosA = (b² + c² − a²)/(2bc). The law works when two sides and the included angle (SAS) or three sides (SSS) are given.
Worked examples
b = 9, c = 4, ∠A = 60°: a² = 81 + 16 − 2·9·4·½ = 97 − 36 = 61, so a = √61.
Sides 4, 7, 9. The angle opposite 9: cos = (16 + 49 − 81)/(2·4·7) = −16/56 = −2/7 < 0, so this angle is obtuse.
Class activity
“Find the side”: each pair picks two sides and an included angle (60°, 90°, 120°) and computes the square of the third side; with 90° they compare with the Pythagorean theorem.
Practice
1
b = 3, c = 5, ∠A = 120°. What is a²?
49
2
b = 4, c = 4, ∠A = 90°. What is a²?
32
3
In a triangle with sides 4, 5, 6, what is the cosine of the angle opposite 6?
1/8
4
Why is the square of the side opposite an obtuse angle greater than the sum of squares of the other two?
cosA < 0, so −2bc·cosA is a positive term.