Post on 14-Jan-2016
Law of Sines
Lesson 6.4
The Law of Sines Let’s now solve oblique triangles
(Δ’s without a right angle):
12 sinac B
BA
C
a
c
bh
12Area sinbc A 1
2 sinab C
sin sin sinbc A ac B ab C
sin sin sinA B C
a b c
When to Use the Law of Sines
When we know one side and two angles (ASA or SAA)
When we know two sides and an angle opposite one of those sides (SSA)
Basically, we need to know a side and the angle opposite that side
B
A =23.5°
C = 112°
a
c
b = 216.75sin 44.5 sin 23.5 sin112
216.75 a c
44.5°
Using the Sine Law
We know 2 sides and angle opposite one of the sides:
Now how would we find angle C and then side c?
A
C
a =9.5
c
b=15
B = 47°
sin 47 sin
15 9.5
A
9 5 47
15
. sinsin A
340 mi
75° 60°
Example 1: A satellite orbiting the earth passes directly overhead and between Phoenix and L.A., 340 mi apart. The
angle of elevation is simultaneously observed to be 60° at Phoenix and 75° at L.A. How far is the satellite from L.A.?
340 mi
75° 60°
sat
LA PHO
060sin
b
cb
45°
0
340
45sin
340 60
45
sin
sinb
Ex 1 Solution: How far is the satellite from L.A.?
32
22
340( )
170 6 mi
416 mi
Height of a Kite Two observers directly under the string
and 30' from each other observe a kite at an angle of 62° and 78°. How high is the kite?
30
78°62°
h
Height of a Kite (cont’d)
30
78°62°
h
The Ambiguous Case (SSA)
Given two sides and an angle opposite one of them, several possibilities exist:
No solution,side too shortto make a triangle
One solution,side equalsaltitude
20°
10 2
20°
10 3.42
The Ambiguous Case (SSA) Two solutions: 2 triangles are possible (why?)
One unique solution,the opposite sideis longer thanadjacent side
20°
105 5
AA'
sin sin 20
10 5
A
Solving for A could give either
an acute or obtuse angle!
Solving for A could give either
an acute or obtuse angle!
20°
10 13.42