Notes Over 8.1Solving Oblique Triangles

To solve an oblique triangle , you need to be given one side, and at least two other parts (sides or angles).

4 cases for oblique triangles

A B

C

ab

c

1. Two angles and any side (AAS or ASA)

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

3. Three sides (SSS)

4. Two sides and their included angle (SAS)

The first two cases use Law of Sines, the last two will use the Law of Cosines.

Notes Over 8.1Law of Sines

If ABC is a triangle with sides a, b, c, then according to the law of sines,

C

c

B

b

A

a

sinsinsin

or

c

C

b

B

a

A sinsinsin

Notes Over 8.1

C

c

A

a

sinsin

o100

o23

12

B

b

A

a

sinsin

o23

o5712

The AAS or ASA CaseSolve ∆ABC

in. 12,57,23 .1 oo aBA

A

B C

o23

o5712

Find: ,C ,b c2357 80

80180 C 100o100

12 b839.0 391.0068.10391.0 b

in. 8.25b

12 c985.0 391.082.11391.0 c

in. 2.30c

Notes Over 8.1The AAS or ASA Case

A

a

B

b

sinsin

o38

o3420

C

c

B

b

sinsin

o34 o10820

Solve ∆ABCmi 20,108,34 .2 oo bCB

A

B Co34 o108

20Find: ,A ,a c10834 142

142180 A 38

o38

20 c951.0 559.002.19559.0 c

mi 0.34c

20 a616.0 559.032.12559.0 a

mi 0.22a

Notes Over 8.1The Ambiguous Case (SSA)

Consider a triangle where you are given a, b, and A

A

abIf the height of the triangle h were equal to a, then it would be a right triangle.

If a < h, then there would be no triangle.

h

b

hAsin Abh sin

a

A is acute

If a > b, then there would be one triangle.

a

If h < a < b, then there would be two triangles.

aa

Notes Over 8.1

B

b

A

a

sinsin

4o2.76o75

C

c

A

a

sinsin o75

42

The SSA Case – One TriangleSolve ∆ABC

in. 2,in. 4,75 .3 o caA B

A Co75

42Find: ,B ,C bo8.28

2sin4 C 965.093.1sin4 C

4825.0sin Co8.28C

8.2875 8.1038.103180 B 2.76

o2.76

4 b971.0 966.0884.3966.0 b

in. 02.4b

Since a > c, one triangle.

Notes Over 8.1The SSA Case – Two Triangles

Solve ∆ABCm 25,m 15,85 .4 o baA

Find h. Abh sin85sin25h

25h 996.0 9.24Since a < h, no triangle.

Notes Over 8.1The SSA Case – Two Triangles

Solve ∆ABCm 31,m 12,5.20 .5 o baA

Find h. Abh sin5.20sin31h

31h 350.0 9.10Since h < a < b, two triangles

1BA

C

o5.20

1231

2B

12

B

b

A

a

sin

sin

3112sin B 350.09.10sin12 B

9047.0sin Bo

1 8.64B

o5.20

12 31

8.641802 B 2.115

o8.64o2.115

5.208.641801 C 7.94 5.202.1151802 C 3.44

1

1

sin

sin C

c

A

ao5.20

127.94

12 1c997.0 350.0m 15.341 c

2

2

sin

sin C

c

A

ao5.20

123.44

12 2c698.0 350.0m 93.232 c

Notes Over 8.1Area of an Oblique Triangle

For any triangle, given two sides and the included angle

BacCabAbc sin2

1sin

2

1sin

2

1Area

6. Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of 102o.

C

o102

m 90a

m 52b 102sin 2

1A 90 52

2m 2289A

Notes Over 8.1

Pg. 588, 8.1 #1-18 all