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