the law of sines!

Objective: Be able to use the Law of Sines to find unknowns in triangles!

Homework: Lesson 12.3/1-10, 12-14, 19, 20

Quiz Friday – 12.1 – 12.3

Quick Review:

What does Soh-Cah-Toa stand for?

What kind of triangles do we use this for?

What if it’s not a right triangle? GASP!! What do we do then??

right triangles

hyp

oppsin hyp

adjcos

adj

opptan

Note:

capital letters always stand for __________!

lower-case letters always stand for ________!

Use the Law of Sines ONLY when:

you DON’T have a right triangle AND

you know an angle and its opposite side

A

B

C

a

b

c

c

C

b

B

a

A sinsinsin

angles

sides

a

sinA

b

sinB

c

sinC

For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

Can also be written in this form:

AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side

If you have 3 dimensions of a triangle and you need to find the other 3 dimensions - they cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation.

Use the Law of Sines if you are given:

29

sin

42

63sin C

Ex. 1: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth.

A

63°

C

a

42

29c

C

b

B

a

A sinsinsin

a

79sin

42

63sin

Csin4263sin29

Csin42

63sin29

C0.38

38˚79˚

79sin4263sina

63sin

79sin42a

3.46a Csin...6152.

Ex. 2: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth.

s

40°

T

r

4.8

89°c

C

b

B

a

A sinsinsin

r

40sin

8.4

51sin

s

89sin

8.4

51sin

40sin8.451sinr

51sin

40sin8.4r

0.4r

51˚

89sin8.451sins

51sin

89sin8.4s

2.6s

Ex. 3: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth.

a. 76,37,7 BmAma 7

37˚

76˚

c

C

b

B

a

A sinsinsin

b

76sin

7

37sin

c

67sin

7

37sin

76sin737sinb

37sin

76sin7b

3.11b

67sin737sinc

37sin

67sin7c

7.10c

67˚

67C

b

c

b. 1.3,70,12 cAma

123.1

70˚

b

1.3

sin

12

70sin C

b

96sin

12

70sin

Csin1270sin1.3

Csin12

70sin1.3

C0.14

96sin1270sinb

70sin

96sin12b

7.12b

14˚

96˚

96B

c

C

b

B

a

A sinsinsin

Csin...2427.

You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c.

* In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter .*


A C

B

70°

80°a = 12c

b

The angles in a ∆ total 180°, so angle C = 30°.

Set up the Law of Sines to find side b:

12

sin 70

b

sin 8012 sin 80 bsin 70

b 12 sin80

sin 7012.6cm

Ex. 3: con’t

Set up the Law of Sines to find side c:

12

sin 70

c

sin 30

12 sin 30 csin70

c 12 sin 30

sin706.4cm

A C

B

70°

80°a = 12c

b = 12.630°

Ex. 3: con’t

Angle C = 30°

Side b = 12.6 cm

Side c = 6.4 cm

A C

B

70°

80°a = 12

c =

6.4

b = 12.630°

Note:

We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations.

Ex. 3: Solution

You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c.

AC

B

115°

30°

a = 30

c

b


AC

B

115°

30°

a = 30

c

b

To solve for the missing sides or angles, we must have an angle and

opposite side to set up the first equation.

We MUST find angle A first because the only side given is side

a.

The angles in a ∆ total 180°, so angle A = 35°.

Ex. 4: con’t

AC

B

115°

30°

a = 30

c

b35°

Set up the Law of Sines to find side b:

30

sin35

b

sin 30

30 sin 30 bsin35

b 30 sin30

sin3526.2cm

Ex. 4: con’t

AC

B

115°

30°

a = 30

c

b = 26.235°

Set up the Law of Sines to find side c:

30

sin35

c

sin115

30 sin115 csin35

c 30 sin115

sin3547.4cm

Ex. 4: con’t

AC

B

115°

30°

a = 30

c = 47.4

b = 26.235°

Angle A = 35°

Side b = 26.2 cm

Side c = 47.4 cm

Note: Use the Law of Sines whenever you are given 2

angles and one side!

Ex. 4: Solution

a

sinA

b

sinB

c

sinC

AAS ASA

Use the Law of Sines to find the missing dimensions of a

triangle when given any combination of these

dimensions.

c

C

b

B

a

A sinsinsin

A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is

Draw a diagram Draw Then find the

Since you know the measures of two angles of the

triangle, and the length of a side

opposite one of the angles you

can use the Law of Sines to find the length of the shadow.

Cross products

Use a calculator.

Law of Sines

Answer: The length of the shadow is about 75.9 feet.

Divide each side by sin

A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water?

Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.

A road slopes 15 above the horizontal, and a vertical telephone pole

stands beside the road. The angle of elevation of the Sun is 65 , and

the pole casts a 15 foot shadow downhill along the road. Fin

d the height

of the pole.

x

15ft15º

65º

B

A

C

Let the height of the pole.

180 90 65 25

65 15 50

sin 25 sin50

1515sin50

27.2sin 25

The height of the pole is about 27.2 feet.

o o o o

o o o

x

BAC

ACB

x

x

We can find the area of a triangle if we are given any two sides of a triangle and the measure of the included angle.

(SAS)

1Area = (side)(side)(sine of included angle)

2

Using two sides and an Angle.

SinBacArea

SinCabArea

SinAbcArea

2

1

2

1

2

1

the law of sines!

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