Transcript 13-5 The Law of Sines (Day 1)

9-5 The Law of Sines
When Drawing Triangles…
Side a is ALWAYS opposite of Angle A, side b is
opposite Angle B, and side c is opposite Angle C.
B
a
C
c
b
A
The Law of Sines:
ABC has sides of length a, b, and c.
Then:
sin A sin B sin C


a
b
c
C
a
b
B
A
c
Ex 1: Solve the triangle, with A = 118°,
C = 36°, and c = 14.
Note: You must have at least one of the ratios in order to
solve the triangle.
sin A sin B sin C


a
b
c C
sin 118 sin 36

a
14
36°
b
21.03  a
a
118°
c = 14
A
B
Ex 1: Solve the triangle, with A = 118°,
C = 36°, and c = 14.
sin A sin B sin C


a
b
c
sin B sin 36

b
14
sin 26 .588

b
14
A+B+C=180
36+B+118=180
B=26
C
21.03
36°
b
118°
A
10 .442  b
26°
c = 14
B
Ex 3: Find the area of triangle ABC
with A=55⁰, c=17, and b=16
1
b = 16
Area  ( side1)( side 2)sin( included Angle )  A
2
55°
c = 17
Note: In order to use this formula
we must know 2 sides and their
included angle.
1
Area  (17 )(16 )sin 55 
2
Area  136.819
Area  111 .405
B
C
Ex 4: There are only 3 cities in the land of Wart.
There is Abicus, Bascilious, and Cabbage.
The 3 cities form a triangle in the magical land. The King’s
magistrate is trying to figure out if there is enough resouces
to build a road from Abicus to Bascilious, as well as from
Cabbage to Abicus. The magistrate knows that he only has
enough resources for 18 km of road for the road from Abicus
to Bascilious. How much more will he need to also complete
the other road? Refer to the given diagram and figure out if
the magistrate has enough resources to do the job.
B
c = 18?
a = 12
122°
C
b
A
Ex 4: Solve the triangle with C = 122°,
a = 12, and c = 18.
Note: We want to verify that a triangle can be created
B
with the given info.
a = 12
cc==18?
18
122°
sin A sin B sin C
C
b
A


We always need at least one ratio to
a
b
c
solve with Law of Sines.
sin A sin 122

sin A  .565
12
18
sin A
A  .601
 .047
A  34 .4 (in degrees)
12
Creating Triangles:
Label each piece of patty paper 1-4 in the top corner.
On piece 1, use your straight edge and draw
any obtuse angle. Make the legs at least an
inch long. Label the angle A, and one of the
legs b.
Lay piece 2 on top of piece 1, draw a leg that is
shorter then leg b. Label that leg a.
Make a triangle out of the 3 segments.
YOU CAN’T!!
Creating Triangles
Lay piece 2 on top of piece 1, extend the length
of leg a until it is the same length as leg b.
Make a triangle out of the 3 segments.
YOU CAN’T!!
Creating Triangles:
Rule 1- If angle A is Obtuse:
a
if a≤b, No triangle
can be created.
b
Note: You can always extend or
shorten this segment; a and b
are the only fixed lengths
a
A
Creating Triangles
Lay piece 3 on top of piece 1, draw a leg that is
longer then leg b. Label that leg a.
How many triangles can you create with the
segments?
One triangle!
Creating Triangles:
Rule 1- If angle A is Obtuse:
a
if a≤b,
No triangle can be
created.
b
A
Rule 2- If angle A is Obtuse:
if a>b,
1 triangle can be
created.
a
b
A
Creating Triangles:
On piece 4, use your straight edge and draw
any acute angle. Make the legs at least an inch
long. Label the angle A, and one of the sides b.
Lay piece 2 on top of piece 1, draw a leg that is
a lot shorter then leg b. Label that leg a. If you
can-use the same leg a you used before.
How many triangles can you make out of the
segments?
YOU CAN’T!!
Creating Triangles:
Rule 3- If angle A is Acute:
a
if bsinA > a, No
triangle can be
created.
b
A
Creating Triangles
Lay piece 3 on top of piece 4, draw a leg that is
longer then leg b. Label that leg a.
How many triangles can you create with the
segments?
One triangle!
Creating Triangles:
Rule 3- If angle A is Acute:
a
if bsinA > a,
No triangle can be
created.
b
A
Rule 4- If angle A is Acute:
if bsinA < a,
1 triangle can be
created.
b
A
a
For the rest of the Rules refer to
pg. 800
Rule 5- If angle A is Acute:
if bsinA < a < b,
2 triangles are created
b
a
a
A
Rule 6- If angle A is Acute:
if bsinA = a,
1 triangle created
b
A
a
Ex 1: Decide whether the given
measurements can form exactly one,
two, or no triangles.
a. A = 140°, a = 5, c = 7
Note: Angle A is opposite side a.
Since A is obtuse, and 5 < 7;
there can be No Triangle created
a=5
c=7
140°
A
b. A = 70°, a = 155, c = 160
Note: Angle A is acute, so try c(sinA)
c(sinA) = 160 (sin70°) = 150.35
150.35 < a = 155 < 160
c(sinA) < a < b, so there can be 2 triangles created
Ex 1: Decide whether the given
measurements can form exactly one,
two, or no triangles.
c. C = 65°, c = 44, b = 32
Note: Angle C is acute, so try b(sinC)
b(sinC) = 32 (sin65°) = 29.002
c =44 > 29.002
c > b(sinC) so there can be 1
triangle created