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Section 9.4:
The Law of Cosines
The Law of Cosines
We use the Law of cosines to solve triangles
that are not right-angled. In particular, when we
know two sides of a triangle and their included
angle, then the Law of Cosines enables us to find
the third side.
Thus if we know sides a and b and their included
angle θ, then the Law of Cosines states:
c² = a² + b² − 2ab cos θ
The Law of Cosines is a extension of the
Pythagorean theorem; because if θ were a right
angle, we would have c² = a² + b².
Example 1.
In triangle DEF, side e = 8 cm, f = 10 cm, and the
angle at D is 60°. Find side d.
d² = e² + f² − 2ef cos 60°
d² = 8² + 10² − 2· 8· 10· ½, since cos 60° = ½,
d² = 164 − 80
d² = 84
d =
Example 2:
In triangle PQR, find side r if side p = 5 in,
q = 10 in, and the included angle of 14°.
r² = 5² + 10² − 2· 5· 10 cos 14°
r² = 25 + 100 – 100 (cos 14°)
r² ≈ 27.97 (store decimal in calc.)
r ≈ 5.3 in.
If we solve the law of cosines for cos C, we
obtain:
cos C = a2 + b2 – c2
2ab
In this form, the law of cosines can be used to
find the measures of the angles of a triangle
when the lengths of three sides are known.
Example 3:
A triangle has sides of lengths 6, 12, and 15.
a. Find the measure of the smallest angle.
b. Find the length of the median to the longest
side.
B
6
A
12
7.5
D
7.5
C
a. The smallest angle of triangle ABC is opposite
the shortest side, AB.
cos C = 122 + 152 – 62
2 · 12 · 15
= 0.925
 C = Cos-1 0.925
≈ 22.3°
b. In triangle BCD, (use the law of cosines)
(BD)2 = 122 + (7.5)2 – 2(12)(7.5)(0.925)
= 33.75
≈5.8
When do you use the Law of Sines and
the Law of Cosines?
Given:
Use:
To find:
SAS
Law of Cosines
The third side and then
one of the remaining
angles.
SSS
Law of Cosines
Any two angles.
ASA or AAS
Law of Sines
The remaining sides.
SSA
Law of Sines
An angle opposite a given
side and then the third
side. (Note that 0, 1, or 2
triangles are possible.)
HOMEWORK
pg. 352 – 353; 1, 2, 5, 6, 9
ASSIGNMENT
pg. 353; 15, 16