Download 6.1: Law of Sines

Review
1.
2.
3.
Solving a right triangle.
Given two sides.
Given one angle and one side.
6.1: Law of Sines
Objectives:
Use the Law of Sines to solve oblique triangles
Find areas of oblique triangles
Use Law of Sines to model & solve real-life
problems
Oblique Triangles
Oblique triangles do not have right
angles.
 Triangles are usually labeled as:

C
a
b
A
c
B
2 Types of Oblique Triangles
All angles are
acute.
One angle is obtuse.
C
C
b
a
b
h
h
A
B
c
a
B
A
c
Must haves for solving Oblique
Triangles
2 angles and any side (AAS or ASA)
 2 sides and an angle opposite one of
them (SSA)
 3 sides (SSS)
 2 sides and their included angle (SAS)
 The first 2 cases can be solved using
the Law of Sines
 The last 2 cases can be solved using
Law of Cosines

What is the Law of Sines?

Follow the directions on the Law of
Sines Discovery notes ( available on
mrtower.wordpress.com )
Law of Sines
If ABC is a triangle with sides a,b,c, then:
a
b
c


sin A sin B sin C
It can also be written as its reciprocal

sin A sin B sin C


a
b
c
2 Angles & 1 Side (AAS)
Given: C=102.3º, B=28.7º, b=27.4 feet
Find: Finishing solving the triangle
1.
C
2.
a
b
A
c
3.
B
4.
Label the givens.
Solve for the missing
angle.
Use the Law of
Sines to find the 2
missing sides.
A = 49 degrees, a =
43.06 ft, c = 55.75 ft
2 Angles & 1 Side (ASA)
Given: A pole tilts toward the sun at an 8º angle
from the vertical, and it casts a 22 foot
shadow. The angle of elevation from the tip of
the shadow to the top of the pole is 43º.
Find: How tall is the pole?
C
b
a
A
c
Label the givens.
Solve for the missing
angle.
Use the Law of Sines
to find the 2 missing
sides.
b = 23.84 ft,
B a = 34.62ft
The Ambiguous Case

am·big·u·ous
 Adjective
/amˈbigyo͞oəs/
– (of language) Open to more than one
interpretation; having a double meaning.
– Unclear or inexact because a choice between
alternatives has not been made.

Synonyms
– equivocal - vague - uncertain - doubtful - obscure
The Ambiguous Case (SSA)
This one is a pain… in the SSA.
Three possible situations:
1. No such triangle exists.
2. Only one such triangle exists.
3. Two distinct triangles can satisfy the
conditions.
Example

Show that there is no triangle for which
a=15, b=25, & A=85°
Label the givens & draw
picture.
2. Use the Law of Sines to
find the missing angle B.
3. Is this result valid? Why or
why not?
4. Invalid since out of Range
5. sinB = 1.66
1.
a
b
h
A
Example
Given: triangle ABC where a=22 inches,
b=12 inches, & A=42°
Find: the remaining side and angles.
4.
Label the givens &
draw picture.
Use the Law of Sines
to find the missing
angle B.
Solve for C.
Solve for c.
5.
B=21o C=117o c=29.29in
1.
C
a
2.
b
A
c
B 3.
Example

1.
2.
3.
4.
5.
6.
7.
Find 2 triangles for which a=12 meters,
b=31 meters and A=20.5°
Label the givens & draw both pictures.
Use the Law of Sines to find the missing
angle B1.
Subtract B1 from 180° to find B2
Subtract the B and A values from 180° to
find C1 and C2.
Use the Law of Sines to find c1 and c2.
Solution 1: B=64.8o C = 94.7o c = 34.15m
Solution 2: B=115.2o C=44.3o c= 23.93m
Area of an Oblique Triangle
1
1
1
Area  bcsin A  absin C  acsin B
2
2
2
The area of an oblique triangle given some
angle  is half the product of the two
adjacent sides and the sine of 
Example

Find the area of a triangular lot having 2
sides of lengths 90 meters and 52
meters and an included angle of 102°
Label the givens & draw picture.
Use the Area Formula to find the area of the lot.
Area = 2,288.82m2
Homework
Check Blackboard.
 Check mrTower.wordpress.com for all
notes, slides, and practice worksheets.
