Chapter 5Analytic

Trigonometry

Pre-Calculus

OHHS

Mr. J. Focht

5.5 Law of Sines

Investigating the Law of Sines–Solving Triangles (AAS, ASA)

–The Ambiguous Case (SSA)

–Applications

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Solving Triangles

• If all triangles were right triangles, we could end the chapter now.

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Law of Sines

• Simple trig and the Pythagorean Theorem won’t work on this triangle.

a

A b

B

cC

B

c

B

b

A

a

sinsinsin

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Solve the triangle.• Find the missing pieces.

A

B

C22

112º

29º

? ?

? • We are given 2 angles and a non-included side.

AAS

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Law of Sines• Solve the triangle.

A

B

C

a

22

c112º

29º

Set up a proportion with 1 missing entry.

112sin29sin

22 a

Solve for a.

29sin

112sin22a

a = 42.142.1

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Law of Sines

• Solve for C• C = 180º - 29º - 112º

= 39ºA

B

C

42.1

22

c112º

29º

39º

• Use Law of Sines

39sin29sin

22 c

29sin

39sin22c = 28.6 28.6

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Now You Try

• Solve this triangle

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Law of Sines - ASA

46

sin 28 sin132

a

20132

46A B

C

ab

C = 180 - 20 - 132

= 28

28

46sin132

sin 28a

72.8a When possible, find the largest side first.

72.8

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Law of Sines - ASA

46

sin 28 sin 20

b

20132

46A B

C

b

28 46sin 20

sin 28b

33.5b

72.8

33.5

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Now You Try

22

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Law of Sines - SSA

• Suppose you were given 2 sides and an angle (non-included) to make a triangle.

Angle A must be across from side a.

Could you do it?

Maybe – it depends on the measurements.

Certain measurements would allow us to make 1 triangle, or 2 triangles, or no triangles.

ba

A

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How Many Solutions?

63º

A

a = 18

b =

25Find the height h=b sin A

h h=25 sin 63º

= 22.3

a is too short to fit.

No Answer!

The height is the shortest side to make the triangle.

22.3

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105º

A

b =

55

How Many Solutions?

a = 73Since a > b, we can form 1, and only 1, triangle.

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72.2º

A

b =

22a = 21

Find All Solutions

First find the height.

h= b sinA

Since h < a, a will fit.

At least 1 triangle can be formed.

But this is not the only way to draw it.

Hint – don’t leave out the lefties

h = 20.95

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72.2º

A

b =

22a = 21

Find All SolutionsUse your left hand to draw it.

h = 20.95

a = 21

Bad news –

You have 2 sets of solutions to solve

72.2º

Ab

= 22

a = 21

h = 20.95

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72.2º

A

b =

22 a = 21

First Solution

Using the Law of Sines

C

B

Bsin

22

2.72sin

21

9.8521

2.72sin22sin 1B

9.219.852.72180C

9.21sin2.72sin

21 cc

2.82.72sin

9.21sin21

c5-5

72.2º

A

b =

22

a = 21

c

Second Solution

B

C

Using the Law of Sines

Bsin

22

2.72sin

21

9.8521

2.72sin22sin 1B

But we can see B is obtuse.

B = 180º - 85.9º = 94.1º

7.131.942.72180C

7.13sin2.72sin

21 c 2.52.72sin

7.13sin21

c5-5

Now You Try

• How many solutions? Do not solve.

1) A = 36, a = 2, b =7

2) C = 36, a =17, c = 16

3) C = 30, a =18, c = 9

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Now You Try

• Find all solutions.

A = 64, a =16, b =17

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Story Problem• A baseball fan is sitting directly behind

home plate in the last row of the upper deck of Comiskey Park in Chicago. The angle of depression to home plate is 29º 54’, and the angle of depression to the pitcher’s mound is 24º 12’. In major league baseball, the distance between home plate and the pitcher’s mound is 60.5 feet. How far is the fan from home plate?

• What is our first step?5-5

Draw a Picture

First find θ

θ=29º54’ - 24º12’ = 5º 42’

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Use Law of Sines

'425sin

5.60

'1224sin

d

d=249.7020342

'425sin

'1224sin5.60

d

d ≈ 249.7

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Now You TrySurveying a Canyon Two markers A and B on the

same side of a canyon rim are 56 ft apart. A third marker C, located across the rim, is positioned so that BAC = 72 and ABC = 53.

(a) Find the distance between C and A. (b) Find the distance between the two canyon

rims. (Assume they are parallel.) 51.9 ft

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Last Word

• What about SSS and SAS?

• They can’t be solved using the Law of Sines.

• We will need to learn something else.

• The Law of Cosines

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Home Work

• P. 484, #2, 6, 10, 14, 16, 18, 20, 22, 24, 25, 26, 30, 40, 47-52

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