Math-3

Lesson 6-5

The Law of Sines

The Ambiguous Case

3. A standard position angle passes

through the point (8, -3). Sin Ө = ?

Quiz 6-4:

2

3

1. Find the measure of angle θ.

2. What is the cosecant ratio for ϴ?

x

25°

15

107°

√5

4. Solve for ‘x’ using the Law of Sines.

C

c

B

b

A

a

sinsinsin

Ө = 33.7°

Csc Ө = 2√5

5

x = 33.9

Sin Ө = -3√7373

A B

a

C

What is the pattern for Law of Sines?

c

b

If three of the four circled

numbers are known,

use law of sines.

b

B

a

A sinsin

B

b

A

a

sinsin

or

Your Turn:

Given the following: solve the triangle.

1. Draw and label the triangle.

2. a = ?

3. C = ?

A = 75º B = 20º b = 20

4. c = ?

Do you remember the 4 triangle congruence theorems?

C

A

15

52°

Angle, (included) side, angle

ASA

48°

B

E

D

1552°

48°

F

b

Triangle ReviewIf the following information is given:

C

c25°

15

107°

Two angles and a “non-included” side

Angle, angle, side AAS

Can you use the

Law of Sines for

AAS?

b

Triangle Review If the following information is given:

C

a

25°

15

107°

Two Angles and the “included side.”

Angle, side, angle ASA

If you have two angles of a triangle, you can find the 3rd angle.

48°

Can you use the

Law of Sines

for ASA?

Two sides and the “included” angle.

12

Triangle Review If the following information is given:

C

a

25°

15

Side, Angle, Side SAS

Law of Sines will NOT work for SAS.

Three sides.

12

Triangle Review If the following information is given:

C

11

15

Side, Side, Side SSS

Law of Sines will NOT work for SSS.

A B

If the following information is given:

An angle and its opposite side and

one other side.

12

B25°

15Side, Side, Angle SSA

Will the Law of Sines work for SSA?

…It depends…

Your Turn:

Can the Law of sines be used for:

SAS ?

SSS ?

ASA ?

no

no

yes, BUT….

AAA? no, trangle can be scaled

up or down in size (no unique triangle).

A B

a

C

Triangle Review:

c

When any combination of

three known values (sides/angles)

are are given,

the possible cases are:

ASA

SAS: Can’t use Law of Sines.

AAS

SSA (the polite order of its letters)

this one presents some

problems for the Law of Sines.

AAA (can be any size, can’t build a specific triangle)

b

SSS: (has no angle given, can’t use Law of Sines)

SSA: The “Ambiguous” CaseIf angle A and opposite side ‘a’, and one other side are

given:

(1) If the adjacent side is longer than the opposite side

AND

(2) If the given angle is “acute”

I can “swing” side ‘a’ until it touches the bottom side at

its end point. This makes another triangle.

A

15 10

Two triangles are possible.

25°

10

SSA: The non-“Ambiguous” Case

If angle A and opposite side ‘a’, and one other side are given:

(1) If the adjacent side is shorter than the opposite side

A

10 15

Only one triangle is possible.

25°

SSA: The non-“Ambiguous” Case

If angle A and opposite side ‘a’, and one other side are given:

(1) If the angle is obtuse

A

1015

Only one triangle is possible.

125°

NOTICE: the opposite side must be longer

than the adjacent side. Why?

SSA CaseIf the given information about a

triangle is:

85

88

A B

C

68º

The angle is not between

the two sides.

This may cause

problems.

A = 68º b = 88 a = 85

1. Draw the triangle.

2. Label the triangle

ba

c

3. Determine

what case it is.

We call this the

ambiguous case.

Ambiguous: Liable to more

than one interpretation.

What is a triangle?

3 segments joined at their endpoints

SSA: The “Ambiguous” Case

IF: the side opposite the given angle is shorter than

the adjacent side, you may have two triangles.

4

25°

5

425°

5

Which of the following cases might

give you two possible triangles?

A = 68º b = 68 a = 85

A = 25º b = 7 a = 5

A = 118º b = 8 a = 20

IF: (1) The given angle is acute (

1. The opposite side is too short to

even make a triangle.

A = 25º b = 7 a = 5

IF: (1) angle is acute (

A = 25º b = 7 a = 5

How can you tell which case it is?

ab

A

b

A

a

bA

a b

A

a

Use the non-opposite side

and the angle to determine the

side length that would give a

right triangle.

How would this help you

to find which “case” it is?

SSA Case:

25°

7 5

y

y = opposite side length for a

right triangle one triangle.

Adjacent > opposite

A = 25º, b = 7, a = 5

Find the opposite side length to make a right triangle.

y = 7 sin 25º

725sin

y

y = 2.95

To make a right angle:

‘a’ must equal 2.95.

5 > “just right” length.

2 triangles.

36.9°

5 4

y

y = opposite side length for a

right triangle one triangle

A = 36.9º, b = 5, a = 4

y = 5 sin 36.9º

59.36sin

y

y = 3

How many triangles are possible?

4 > “just right” length.

2 triangles.

SSA Case5

No triangle !!!

Side opposite the angle

is shorter than the shortest

possible opposite side.

A B

C

36.9º

2

3

Summary (ambiguous case)

A’s & S’s #of Triangles:

AAS

Given Angle and Sides

Adj > opp; opp > “just right”2

SSA

1

AAA

0

1

SSA

Adjacent side < opposite side

Adj > opp; opp = “just right”

Adj > opp; opp < “just right”

angle

adj opp

A = 52º, a = 32, b = 42

How many triangles? (one, two, or none)

A = 28º, C = 75º, c = 20

A = 40º, a = 13, b = 16

(Hint: draw the triangle!!!)

Solve the Triangle:

Opp side > right angle side ?

4

B

5

20°

520sin

y

A = 20º, a = 4, b = 5

y

20sin5y 71.1

1) Draw the triangle (ambiguous SSA Case?) (adj > opp) YES!

2) How many triangles? find opposite side for a right triangle.

2 triangles

Solve the Triangle:If it is two triangles, the law of sines will give the acute angle:

54

B

5

20° 20°

A = 20º, a = 4, b = 5

44

12

4

20sin

5

sin 1

4

20sin5sin 1

1

4275.0sin 11

3.251

4275.0sin 1

C

c

3.2520180 Cm 7.134

Use law of sines to find ‘c’.

134.7°

Note: Do you see the Blue Isosceles Triangle?

4

B

5

20°

23.25180

25.3°

4

25.3°2

Both “base angles”

are congruent.

= 154.7°

Note: Do you see the

linear pair of angles?

154.7°20°

54

C

c

Solve the 2nd triangle.

7.15420180 Cm

= 5.3°

Use Law of Sines to solve for ‘c’.

Finding the Height of a Pole

Two people are 2.5 meters apart on opposite sides of a pole.

The angles of elevation from the observers to the top of the

pole are 51° and 68°. Find the height of the pole.

51º 68º

b a

1. Find either ‘a’ or ‘b’ using

Law of Sines.

2. Solve the right triangle

using right triangle rules

where height is the side

opposite the angle.2.5