Chapter 7 – Applications of Trigonometry

7 Days

7.1 Law of SinesTwo days

In any triangle, the ratio of the sine of any angle to the side opposite that angle is equal to the ratio of the sine of another angle to the side opposite that angle.

Law of Sines

Law of Sines

C B

A

cb

a

c

C

b

B

a

A sinsinsin

The Law of Sines can be used anytime that we are given the following information about a triangle:

◦ 1. Two sides and an angle opposite one of them (SSA)

◦ 2. Two angles and any side (AAS or ASA)

If we have either SAS or SSS the Law of Sines will not be sufficient. (to be continued…)

Law of Sines

Find the missing pieces of the triangle given:

Examples

C B

A

cb

a

20,45,120 cBC

Find the missing pieces of the triangle given:

Examples

C B

A

cb

a

115,50,82 cbB

Find the missing pieces of the triangle given:

What happened in the previous example…

??

??

50

115

82*

115,50,82 cbB

If we are given two angles and any side, the Law of Sines results in exactly one triangle.

However, if we are given two sides and an angle opposite one of the those sides, we could get one triangle, two triangles, or no triangles.

Things to consider…

Find the missing pieces of the triangle given:

Examples

C B

A

cb

a

115,140,'2053 ca

A surveyor is trying to determine the distance between A and B and chooses a point C that is 375yds from A and 530yds from B. If <BAC has a measure of , what is the distance between A and B?

Applications

'3049

p527 #1,2,4,5,14,17 - 19,21

Homework

7.2 Law of CosinesFour Days

Solve using the Law of Sines.

Can we solve with the Law of Sines?

Example

20b ,18 ,15 cA

18c ,15 ,20 ba

We need the Law of Cosines to solve triangles that are SAS or SSS.

Law of Cosines

bc

cbaA

Abccba

2cos

cos2

:Cosines of Law The

222

222

SAS -

Example 1

15b ,18 ,48 aC

SSS – Note: Find the largest angle first since arccos can

give obtuse angles!

Example 2

36c ,19 ,25 ba

p536 #1,3,4,6,9,11,13 - 15

Homework

A reconnaissance airplane P, flying at 10,000ft above point R on the surface of the water, spots a submarine S at an angle of depression of 37* and a tanker T at an angle of depression of 21*, as shown in the diagram. If <SPT is 110*, what is the distance between the tanker and the submarine.

Law of Cosines – Day 2

p528 #24 p537 #12 p580 #41,43,45,47

Homework

7.2 Heron’s FormulaOne Day

We can find the area of a triangle given the lengths of all three sides or two sides and an angle.

If we only know the lengths of sides we will use Heron’s Formula:

Case 1 - SSS

perimeter the ;2

where

))()((

21cba

s

csbsassArea

Find the area of the triangle if a=47, b=58, and c=78

Example

Find the area of a triangle given A=100*, b=16, and c=18.

Case 2 – Two sides and an angle

=100

c=18

b=16

a=h=_____

=100

c=18

b=16

a=

100sin18

h

100sin18h

bhA 21

)100sin18)(16(21 A

8.141A

p537 #18,22,29,31,33,35

Homework

5-8 paper Glencoe area of triangle

Homework

7.1 & 7.2 review paper

Homework