Lesson 6.1- Law of Sines

Provided by Vivian Skumpija and Amy Gimpel

When to use the Law of Sines

• To solve an oblique triangle• Given:– Two angles and any side (AAS or ASA)– Two sides and an angle opposite one of them

(SSA or ASS)

***Always remember to keep your calculator in degree mode ***

AAS• If C=102.3 °, B=28.7°, b=27.4 • A=180-102.3-28.7 49°Since you have an angle measurement and its

opposite side you can set up the equation…

a= 43.06

c=57.75

75.577.28sin

4.27

3.102sin

06.437.28sin

4.27

49sin

c

a

NOTE: To solve for c, you always go back to your original equation to avoid using rounded values.

ASS (Single Solution)

• If b= 795.1, c=775.6, B= 51.85°

Since side b is greater than the other given side, there is only one solution

51.85 °

B

AC

775.6

795.1

19.98906.78sin85.51sin

1.795

06.7809.5085.51180

09.50sin

6.775

85.51sin

1.795

a

A

c

989.19

78.06 ° 50.09 °

NOTE: To solve for sin c, you must use the answer with inverse sine (on calculator: 2nd Sin, 2nd ANS)

ASS (No Solution)

• a=7, b=15, A=98°

This triangle can not be solved for any answers, which would make it NO SOLUTION.

ERRORDOMAINb

sin

15

98sin

7

a’=10

B’

A’52© 102.3 ©

25.7 ©

b’=5.5

c’=12.4

ASS (Two Solutions)

• First: solve for the regular triangle

A

B

C

52 ©

a=10c=12.4

b=9.8

C’

7.77sin

4.12

52sin

10

C

8.93.50sin52sin

10

b

5.57.25sin52sin

10

3.1027.77180'

b

C

77.7 ©

3.507.7752180 B

10

77.7 ©

Because the side across from the given angle is smaller than the other side, there is another set of solutions…

Finding the Area

• Since sides a and b are given and so is angle C…

BacCabAbcArea sin2

1sin

2

1sin

2

1

102°

a=90

b=52

sqmeters2289102sin52902

1

Lesson 6.2- Law of Cosines

When to use the Law of Cosines

• To solve an oblique triangle• Given:– Three sides (SSS)– Two sides and their included angle (SAS)

***Always remember to keep your calculator in degree mode ***

• a=3, b=5, c=7 pick the largest

solve for the rest of

the triangle

SSS

120

2

1cos

cos2

130

cos3015

cos303449

cos3025949

cos532537

1

222

C

C

c

c

c

c

c

21.38sin

5

120sin

7

79.21sin

3

120sin

7

b

a

SAS• a=4.2, c=7.5, B=32°Because you have angle B, you want to solve for

side b…

…Then you solve for the rest using the Law of Sines

52.4

46.20

46.20

43.5389.73

32cos5.72.425.72.4

2

2

2

222

b

b

b

b

b

5.29sin

2.4

32sin

52.4

a

Heron’s Area Formula

If a=43m, b=53 m, c=72 m

289.1131)12)(31)(41(84

)7284)(5384)(4384(84

842/725343

meters

2/))()((

cbawhereS

csbsass

Substitute for S

THAT’S ALL THERE IS TO

IT