7.3 Day 1 The Law of Cosines May 04, 2009
7.3 The Law of Cosines
Day 1
Objective: Use the law of cosines to find measures of sides and angles in triangles.
Apr 19 ­ 9:19 AM
1
7.3 Day 1 The Law of Cosines May 04, 2009
With a right triangle, you can use the trig identities to find missing sides or angles.
tan =
opp
adj
cos
=
adj
hyp
sin
opp
=
hyp
hyp
opp
adj
Apr 18­6:41 PM
2
7.3 Day 1 The Law of Cosines May 04, 2009
But what about a triangle that does not have a right angle
and can NOT be solved using the Law of Sines?
We need another strategy to find missing sides and angles.
Apr 18­6:44 PM
3
7.3 Day 1 The Law of Cosines May 04, 2009
This is where the Law of Cosines comes in...
a
c
β
γ
α
b
Start with any triangle that does not have a right angle.
Notice: The ANGLES are labeled with GREEK LETTERS. We will always label triangles so a is opposite α, b is opposite β, and c is opposite γ.
Apr 18­6:49 PM
4
7.3 Day 1 The Law of Cosines May 04, 2009
Law of Cosines: For any triangle with sides a, b, c and opposite angles α, β, γ, respectively, the following is true. When given any 2 sides and the included angle this form of the formula is used.
α
b
γ
c
β
c2 = b2 + a2 ­ 2ab cos γ
b2 = a2 + c2 ­ 2ac cos β
a
a2 = b2 + c2 ­ 2bc cos α
Apr 20 ­ 3:03 PM
5
7.3 Day 1 The Law of Cosines May 04, 2009
Law of Cosines: When given all 3 sides, you can find an angle by rearranging the formula.
b2 = a2 + c2 ­ 2ac cos β
b2 ­ a2 ­ c2 = ­ 2ac cos β
b
α
γ
a
c
β
b2 ­ a2 ­ c2 = cos β
­2ac
­b2 + a2 + c2 = cos β
2ac
β = cos
­1
α = cos
γ = cos
­1
­1
(
(
(
a2 + c2 ­ b2
2ac
b2 + c2 ­ a2
2bc
a2 + b2 ­ c2
2ab
(
(
(
Apr 20 ­ 3:03 PM
6
7.3 Day 1 The Law of Cosines May 04, 2009
Example #1:
b = 3.7
a2 = b2 + c2 ­ 2bc cos α
c = 4.3
a2 = (3.7)2 + (4.3)2 ­ 2(3.7)(4.3)cos(39o)
o
α = 39
a = 2.7
Solve the triangle.
α
b
γ
c
β
a
o
β = 59.6
o
γ = 81.4
Apr 20 ­ 3:29 PM
7
7.3 Day 1 The Law of Cosines Example #2:
May 04, 2009
c2 = b2 + a2 ­ 2ab cos γ
a = 5.1
b = 4.6
o
γ = 47
c = 3.9
Solve the triangle.
α
b
γ
c
β
a
o
α = 73
o
β = 60
Apr 20 ­ 3:29 PM
8
7.3 Day 1 The Law of Cosines Example #3:
a = 4.1
May 04, 2009
b2 + c2 ­ a2
cos α = 2bc
b = 6.4
c = 8.6
Solve the triangle.
α
b
γ
o
α = 27
c
β
a
β = 45.1
o
γ = 107.9
o
Apr 20 ­ 3:29 PM
9
7.3 Day 1 The Law of Cosines May 04, 2009
page 546 (10 ­ 30 even)
skip #s 12 & 18 Apr 20 ­ 3:56 PM
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