1

What you will learn

How to solve triangles by using the Law of Sines if the measures of two angles and a side are given

How to find the area of a triangle if the measures of two sides and the included angle or the measures of two angles and a side are given

Objective: 5-6 Law of Sines

2Introduction

Over the next three sections, we are going to learn to solve triangles that are not right triangles.

- Law of Sines (including a “special” case)

- Law of Cosines

Objective: 5-6 Law of Sines

3

A Little Background

CD

A

B

D C

How do you know?

Therefore, angleD and angle C willhave the same sine

2r

C

cr

r

cC

r

cD

sin2

2sin

2sin

CD

Objective: 5-6 Law of Sines

4

The Law of Sines

The Law of Sines: Let triangle ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C respectively.

What we are learning today applies to the following congruencies: ASA or AAS

C

c

B

b

A

a

sinsinsin

Objective: 5-6 Law of Sines

5

Example Solve triangle ABC if A = 33o, B = 104o

and b = 37.9

A

B

CB=37.9

Objective: 5-6 Law of Sines

6

You Try Solve triangle LMN if L = 29o, M=112o,

andl=22.

L

M

N

Objective: 5-6 Law of Sines

7

The Inevitable Word Problem A baseball fan is sitting directly behind home plat in the last

row of the upper deck of U.S. Cellular Field in Chicago. The angle of depression to home plate is 29o54’, and the angle of depression to the pitcher’s mound is 24o12’. 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? Remember, the angle of elevation is congruent to the angle of depression (they are alternate interior angles).

60.5 ft.

24o12

24o12

d

Objective: 5-6 Law of Sines

8

Finding Area of Triangles When 2 Sides are Known

Let triangle ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, and C, respectively. Then the area, K, can be determined using one of the following formulas.

K = ½ bc sin A K = ½ ac sin BK = ½ ab sin C

Objective: 5-6 Law of Sines

9

Example Find the area of triangle ABC if a = 4.7, c

= 12.4, and B = 47o20’

AB

C

4.7

12.4

47o20’

Objective: 5-6 Law of Sines

10

You Try Find the area of triangle ABC if b = 21.2,

c = 16.5, and A = 25o.

Objective: 5-6 Law of Sines

11

Area if Two Angles and One Side are Known

Let triangle ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, and C respectively. Then the area K can be determined using one of the following formulas.

A

CBaK

sin

sinsin

2

1 2B

CAbK

sin

sinsin

2

1 2

C

BAcK

sin

sinsin

2

1 2

Objective: 5-6 Law of Sines

12

Example

Find the area of triangle DEF if d=13.9, D = 34.4o, and E = 14.8o

DE

F

13.9

Objective: 5-6 Law of Sines

13

You Try Find the area of triangle JKL if j = 45.7, K

= 111.1o, and L = 27.3o

Objective: 5-6 Law of Sines

14

Homework page 316, 11, 13, 17-23 odd, 25, 28