7.2 Day 2 The Law of Sines May 01, 2009
7.2 The Law of Sines
Day 2
Objective - Use the law of sines to
find measures of sides and angles
in ambiguous case triangles.
Apr 26­6:41 PM
1
7.2 Day 2 The Law of Sines May 01, 2009
Find the following sine values:
sin35o = 0.574
sin145o = 0.574
sin50o = 0.766
sin130o = 0.766
sin166o = 0.242
sin14o = 0.242
What pattern do you notice? Apr 26­6:43 PM
2
7.2 Day 2 The Law of Sines May 01, 2009
The Ambiguous Case: When two sides and the opposite angle of one of the sides is known, the information may result in one triangle, two triangles, or no triangle at all.
No Triangle
a < h
b
One Right Triangle
a = h
b
a h
a h
α
α
One Triangle
a > b
Two Triangles
a < b and a > h
b
a
a
h
h
b
a
α
α
Apr 26­6:59 PM
3
7.2 Day 2 The Law of Sines May 01, 2009
Example #1: Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
3
a = 3, b = 2, α = 40o
c
β
40o
γ
ONE TRIANGLE
2
β1 = 25.4o
o
o
o
γ = 180 ­ (25.4 + 40 )
o
o
γ = 180 ­ 65.4
o
γ = 114.6
sinβ
sin40o
2 = 3
o
2sin40
sinβ =
3
sinβ ≈ 0.43
3
c
=
sin40o
sin114.6o
β ≈ sin­1(0.43)
o
o
β1 ≈ 25.4 and β2 ≈ 180 ­ 25.4
β2 ≈ 154.6o
o
0
= 4
o
!
3sin114.6o
c = sin40o
c ≈ 4.24 e α
us o
a
ec .6
e b 194 gle!
l
b
n
si o =
os .6 a tria
p
4
T 5 or NO o + 1 IG f
40 o B
To
Apr 26­6:44 PM
4
7.2 Day 2 The Law of Sines May 01, 2009
Example #2: Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
a = 6, b = 8, α = 35o
c
β
6
35o
γ
8
sinβ
sin35o
8 = 6
o
8sin35
sinβ =
6
sinβ ≈ 0.76
TWO TRIANGLES
WORK ON NEXT PAGE!
β ≈ sin­1(0.76)
β1 ≈ 49.9o and β2 ≈ 180o ­ 49.9o
β2 ≈ 130.1o
.
o
!! .1 r
les 165 n a t
g
ian o = k i
Tr 0.1 wor
O 13 s TWo + ngle
35 th a
Bo
gle
ian
Apr 26­6:44 PM
5
7.2 Day 2 The Law of Sines May 01, 2009
Example #2: Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
c
β
6
35o
γ
a = 6, b = 8, α = 35o
8
TWO TRIANGLES
o
β2 ≈ 130.1o
o
o
o
γ = 180 ­ (130.1 + 35 )
o
o
γ = 180 ­ 165.1
o
γ = 14.9
β1 ≈ 49.9 o
o
o
γ = 180 ­ (49.9 + 35 )
o
o
γ = 180 ­ 84.9
o
γ = 95.1
c 6
= sin35o
sin95.1o c 6
= sin35o
sin14.9o 6sin95.1o c =
sin35o
6sin14.9o c =
sin35o
c ≈ 10.42
c ≈ 2.69
Apr 26­6:44 PM
6
7.2 Day 2 The Law of Sines May 01, 2009
Example #3: Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
2
a = 2, c = 1, γ = 50o
1
β
50o
α
b
sin50o
1
2sin50o
1
sinα ≈ 1.53
sinα
=
2
sinα =
NO TRIANGLE!!
­1
α ≈ sin (1.53)
­1
α ≠ sin (1.53)
ch .
hi gle
r w an
fo tri
E! α no GL gle is AN an ere
RI no th
T is so NO ere > 1
Th n α si
Apr 26­6:44 PM
7
7.2 Day 2 The Law of Sines May 01, 2009
Homework: page 539 (26 ­ 36 even, 41, 47, 50)
Apr 26­8:05 PM
8