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Transcript SINCOSonline

There is no doubt that
the 3 PTRs are
extremely useful when
solving problems
modeled on a right
triangle.
Unfortunately, the world
does not consist only of
right triangles…
As a matter of fact, right
triangles end up being
more of a rarity than
commonplace.
There are many
situations where
angles other than
90O are present.
Does that mean when we
come across a situation
that can only be modeled
with a non-right triangle
that we abandon our
pursuit?….
No Way!!!!
There exists two Laws of
Trigonometry that allow one to
solve problems that involve nonright Triangles:
Remember
A capital letter represents an
angle in a triangle, and a small
letter represents a side of a
triangle
a
A
If b,c and O are all known, then O
is called a “Contained Angle”
(the blue line also forms a “c”, [kind of] which
is how I remember to use the “c”osine law
in this case..)
C
a
b
A
O
c
B
The Cosine Law can be used to
find the length of the opposite
side to O
In this case, the length of side a
C
a
b
A
O
c
B
In General:
a2 = b2 + c2 – 2bcCosOo
C
a
b
A
O
c
B
For Example: Find a
C
a
8m
50o
A
10m
B
a2 = b2 + c2 – 2(b)(c)CosAo
a2 = 82 + 102 – 2(8)(10)Cos50o
a2 = 61.15m You should be able to
load this into your
a = 7.8 m
calculator directly
C
from left to right…if
not, see me
a
8m
50o
A
10m
B
The Sine Law
If the triangle being solved
does not consists of a right
triangle (3PTRs) or a
contained angle (Cosine
Law), then another tool
must be used.
If a corresponding angle and
side are known, they form an
“opposing pair”
C
b
A
a
O2
O1
c
B
The Sine Law can be used to
determine an unknown side or
angle given an “opposing pair”
C
b
A
a
O2
O1
c
B
The Sine Law
SinA = SinB = SinC
a
b
c
C
a
b
A
c
B
Find the length of a
C
57o
We can not use the
Cosine Law because
there is not a
contained angle…
a We must therefore
look for an opposite
pair. Hmmm…..
A
73o
A-HA!!!
c
24
N
(it’s all good)
Find the length of a
C
a =
Sin73o
57o
a = 27.4
a
A
73o
c
24
24
Sin57o
Again, this can be
put directly into
your calculator.
See me for help.
N
Pg 290
1a,c,d,e
4a,c,e
5a,c
6
8,10,12,14
Pg 295
1
(11 unco,
stop here)
The Ambiguous
Case
Find A
SinA =
11
11
9
48o
A
Sin48o
9
A = 65.3o
Does that make
sense?
No Way!!!
Side 9 can also be drawn as:
11
48o
Could A be
o
65 in this
case?
9
A
This type of discrepancy is
called the “Ambiguous Case”
Be sure to check the diagram
to see which answer fits:
o
O, or 180 - O