WHAT YOU WILL LEARN:• USE THE LAW OF S INES TO SOLVE OBLIQUE

TRIANGLES WHEN YOU KNOW TWO ANGLES AND ONE S IDE (AAS OR ASA) .

• USE THE LAW OF S INES TO SOLVE OBLIQUE TRIANGLES WHEN YOU KNOW TWO SIDES AND THE ANGLE OPPOSITE ONE OF THEM ( SSA) .

Law of Sines

Introduction

In this section, we will solve oblique triangles – triangles that have no right angles.

As standard notation, the angles of a triangle are labeled A, B, and C, and their opposite sides are labeled a, b, and c.

To solve an oblique triangle, we need to know the measure of at least one side and any two other measures of the triangle—either two sides, two angles, or one angle and one side.

Introduction

The Law of Sines can also be written in the reciprocal form:

Given Two Angles and One Side – AAS

For the triangle below C = 102, B = 29, and b = 28 feet. Find the remaining angle and sides.

By the triangle angle-sum theorem, A = 49.

Law of Sines

Try this:

A

B

C

ca

b

mmcBA oo 45,67,43

By the triangle angle-sum theorem, C = 70.

𝑎=𝑐

𝑠𝑖𝑛𝐶 ( 𝑠𝑖𝑛𝐴 )= 45𝑠𝑖𝑛70 °

( 𝑠𝑖𝑛 43 ° )≈32.66 𝑓𝑒𝑒𝑡

𝑏=𝑐

𝑠𝑖𝑛𝐶 (𝑠𝑖𝑛𝐵 )= 45𝑠𝑖𝑛70 °

( 𝑠𝑖𝑛67 ° )≈44.08 𝑓𝑒𝑒𝑡

Let’s look at this: Example 1

Given a triangle, demonstrate using the Law of Sines that it is a valid triangle (numbers are rounded so they may be up to a tenth off):a = 5 A = 40o

b = 7 B = 64.1o

c = 7.55 C = 75.9o

Is it valid??

𝑎𝑠𝑖𝑛𝐴=

𝑏𝑠𝑖𝑛𝐵=

𝑐𝑠𝑖𝑛𝐶

???

, so YES

a = 5 A = 40o

b = 7 B = 115.9o

c = 3.175 C = 24.1o

Is it valid??

𝑎𝑠𝑖𝑛𝐴=

𝑏𝑠𝑖𝑛𝐵=

𝑐𝑠𝑖𝑛𝐶

???

, so YES

Why does this work? In both cases a, b and A are the same (two sides and an angle) but they produced two different triangles. Why??

Here is what happened

What is the relationship between 64.1and 115.9?

Remember the sine of an angle in the first quadrant (acute: 0o – 90o) and second quadrant,(obtuse: 90o – 180o)are the same!

The Ambiguous Case (SSA)

The Ambiguous Case (SSA)

In our first example we saw that two angles and one side determine a unique triangle.

However, if two sides and one opposite angle are given, three possible situations can occur:

(1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles may satisfy the

conditions.

The Ambiguous Case (SSA)

Example – Single-Solution Case—SSA

In triangle ABC, a = 12 inches, b = 5 inches, and A = 31. Find the remaining side and angles.One solution: a b

aA

bB sinsin

aAbB sinsin

1231sin5sin B

39.12B B is acute.

Now, you can determine that C 180 – 31 – 12.39 = 136.61.

Then, the remaining side is

Aa

Cc

sinsin C

Aac sin

sin

)61.136sin()31sin(

12

c

inches01.16

Example – No-Solution Case—SSA

In triangle ABC, a = 4 inches, b = 14 inches, and A = 60. Find the remaining side and angles.

aA

bB sinsin

aAbB sinsin

03.3sin B

NO SOLUTION

Example – Two-Solution Case—SSA

In triangle ABC, a = 4.5 inches, b = 5 inches, and A = 58. Find the remaining side and angles.One solution: a b

aA

bB sinsin

aAbB sinsin

5.458sin5sin B

56.109 ,44.70 21 BB

Two solutions

Now, you can determine that C1 180 – 58 – 70.44 = 51.56

C2 180 – 58 – 109.56 = 12.44

Aa

Cc

sinsin 11 sin

sinC

Aac

inchesc 17.4)56.51sin()58sin(

5.41

inchesc 14.1)44.12sin()58sin(

5.42

22 sinsin

CAac

WHAT YOU WILL LEARN:• USE THE LAW OF COSINES TO SOLVE OBLIQUE

TRIANGLES WHEN YOU KNOW THREE S IDES (SSS) .• USE THE LAW OF COSINES TO SOLVE OBLIQUE

TRIANGLES WHEN YOU KNOW TWO SIDES AND THE INCLUDED ANGLE (SAS) .

Law of Cosines

CabCosbac

BacCoscab

AbcCoscba

2

2

2

222

222

222

abcbaC

acbcaB

bcacbA

2cos

2cos

2cos

222

222

222

Always solve for the angle across from the longest side first!

Example – SSS

In triangle ABC, a = 6, b = 8, and c = 12. Find the three angles.

abcbaC

acbcaB

bcacbA

2cos

2cos

2cos

222

222

222

28.117)4583.(cos

4583.8621286cos

1

222

C

C

aA

cC sinsin

cCaA sinsin

34.36)38.2628.117180(38.26)4444.0(sin

4444.012

28.117sin6sin

1

BA

A

Example – SAS

In triangle ABC, A = 80, b = 16, and c = 12. Find the remaining side and angles.

aA

bB sinsin

aAbB sinsin

36.20)36.12080180(36.40)64.5980180(

36.120or 64.59)8629.0(sin

8629.026.1880sin16sin

1

CCB

B

CabCosbac

BacCoscab

AbcCoscba

2

2

2

222

222

222

26.18

)80cos(121621216 222

aa