MTH 112: Elementary Functions
1/13
MTH 112: Elementary Functions
Section 8.2: Law of Cosines
Derive the Law of Cosines.
Solve triangles.
Find areas of triangles.
Boris Iskra
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
2/13
Solving oblique triangles
Solving Oblique triangles
SSS (side side side)
SAS (side angle side). Two sides and the angle between are given.
C
?
b
A
Boris Iskra
C
a
?
?
c
?
b
B A
?
?
α
c
B
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
3/13
Solving oblique triangles
SAS (side angle side). Two sides and the angle between are given.
C
b
A
?
c
?
?
?
α
C
C
B A
?
a
β
?
c
γ
b
B A
a
?
?
?
B
They are all the same!!
Boris Iskra
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
4/13
Law of Cosines
Any triangle with standard labeling satisfies:
Law of Cosines
a2 = b2 + c2 − 2bc cos(α)
b2 = a2 + c2 − 2ac cos(β )
c2 = a2 + b2 − 2ab cos(γ)
C
b
A
Boris Iskra
?
c
?
?
?
α
C
C
B A
?
a
β
?
c
γ
b
B A
a
?
?
?
B
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
5/13
Example (SSS)
If a = 7, b = 5, and c = 8, solve triangle ABC.
First, find α. Angle α can be found
using the law of cosines.
C
γ
5
7
B
Boris Iskra
α
β
8
A
α = 60◦
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
6/13
Example (SSS) continued
If a = 7, b = 5, and c = 8, solve triangle ABC.
Second, find β . Angle β can be found
using the law of cosines.
C
b2 = a2 + c2 − 2ac cos(β )
γ
5
7
B
Boris Iskra
60◦
β
8
A
β ≈ 38.21◦
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
7/13
Example (SSS) continued
If a = 7, b = 5, and c = 8, solve triangle ABC.
Finally, find γ. Angle γ can be found
using the law of cosines.
C
c2 = a2 + b2 − 2ab cos(γ)
γ
5
7
B
60◦
38.21◦
8
A
NOTE: It’s easier to find γ using the fact that the sum of angles in any
triangle is 180◦ .
Boris Iskra
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
8/13
Example (SAS)
If b = 5, c = 7, and α = 138◦ , solve triangle ABC.
First, find a. Size a can be found
using the law of cosines.
C
a2 = b2 + c2 − 2bc cos(α)
γ
a
5
A
Boris Iskra
138◦
β
7
B
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
9/13
Example (SAS) continued
If b = 5, c = 7, and α = 138◦ , solve triangle ABC.
Second, find β . Angle β can be found
using the law of cosines.
C
b2 = a2 + c2 − 2ac cos(β )
γ
11.2
3
5
A
138◦
β
7
B
γ≈
Boris Iskra
≈ 17.34◦
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
10/13
Area of a triangle
Any triangle with standard labeling satisfies:
C
γ
b
A
a
β
α
c
B
Area of a triangle
1
1
1
Area of triangle = ab sin(γ) = ac sin(β ) = bc sin(α)
2
2
2
Boris Iskra
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
11/13
Example
If b = 18, c = 23, and α = 58◦ . Find the area of the triangle.
23
1
Area = 18 · 23 · sin(58◦ )
2
≈ 411.0
58◦
18
Boris Iskra
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
12/13
Area of a triangle
For any triangle.
b
a
c
Heron’s formula
Area of triangle =
p
s(s − a)(s − b)(s − c)
1
where s = (a + b + c)
2
Boris Iskra
Department of Mathematics. Oregon State University
MTH 112: Elementary Functions
8.2: Law of Cosines.
13/13
Example
If a = 12, b = 18 and c = 23. Find the area of the triangle.
12
23
Area
18
Boris Iskra
≈ 106.9179
Department of Mathematics. Oregon State University