Law of Sines

Lesson 6.1

2

Working with Non-right Triangles

We wish to solve triangles which are not right triangles

BA

C

a

c

bh

sin sin

sin sin

sin sin

sin sin

h hA B

b ab A h a B

b A a B

A B

a b

sinC

c

3

Using the Sine Law

If we know two angles and one side, we can solve the triangle Actually, if we know two angles, we know

all three

BA =23.5°

C = 112°

a

c

b = 216.75 180 - 23.5 - 112

44.5

B

sin 44.5 sin 23.5 sin112

216.75 a c

4

Using the Sine Law

If we know two sides and an opposite angle We can solve the whole triangle

Now how to find angle C and then side c?

A

C

a =9.5

c

b=15

B = 47°

sin 47 sin

15 9.5

A

5

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 1

20°

10 3.42

6

The Ambiguous Case (SSA)

Two possible triangle could result (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

7

Try It Out

Solve these triangles – watch for ambiguous case

28°78°

44

32°

9.0

14

8

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°

?