Work out problems on board

1

• Work out problems on board
• Add visuals (ranges of arccos and arcsin) to show
  why you use LOC for big angles and LOS for small
  angles

2
6.2 The Law of Cosines
3
Which proved triangles congruent in Geometry?

• SSS
• ASA
• AAS
• SAS
• AAA
• ASS

4
The same ones that define a specific triangle!

• SSS - congruent
• ASA - congruent
• AAS congruent
• SAS congruent
• AAA not congruent
• ASS not congruent

5
Which proved triangles congruent in Geometry?

• SSS - congruent
• ASA congruent Solve w/ Law of Sines
• AAS congruent Solve w/ Law of Sines
• SAS congruent
• AAA not congruent
• ASS not congruent

6
Solving an SAS Triangle

• The Law of Sines was good for
• ASA - two angles and the included side
• AAS - two angles and any side
• SSA - two sides and an opposite angle (being
  aware of possible ambiguity)
• Why would the Law of Sines not work for an SAS
  triangle?

15
26
No side opposite from any angle to get the ratio
12.5
7
Let's consider types of triangles with the three
pieces of information shown below.
We can't use the Law of Sines on these because we
don't have an angle and a side opposite it. We
need another method for SAS and SSS triangles.
SAS
AAA
You may have a side, an angle, and then another
side
You may have all three angles.
AAA
This case doesn't determine a triangle because
similar triangles have the same angles and shape
but "blown up" or "shrunk down"
SSS
You may have all three sides
8
Do you see a pattern?
9
Deriving the Law of Cosines

• Write an equation using Pythagorean theorem for
  shaded triangle that
• only includes sides and angles of the
• oblique triangle.

C
b h a
k c - k
A B
c
10
Since the Law of Cosines is more involved than
the Law of Sines, when you see a triangle to
solve you first look to see if you have an angle
(or can find one) and a side opposite it. You
can do this for ASA, AAS and SSA. In these cases
you'd solve using the Law of Sines. However, if
the 3 pieces of info you know don't include an
angle and side opposite it, you must use the Law
of Cosines. These would be for SAS and SSS
(remember you can't solve for AAA).
Since the Law of Cosines is more involved than
the Law of Sines, when you see a triangle to
solve you first look to see if you have an angle
(or can find one) and a side opposite it. You
can do this for ASA, AAS and SSA. In these cases
you'd solve using the Law of Sines. However, if
the 3 pieces of info you know don't include an
angle and side opposite it, you must use the Law
of Cosines. These would be for SAS and SSS
(remember you can't solve for AAA).
11
Ex. 1 Solve a triangle where b 1, c 3 and A
80
Draw a picture.
B
This is SAS
3
a
Do we know an angle and side opposite it? No so
we must use Law of Cosines.
C
80
1
Hint we will be solving for the side opposite
the angle we know.
times the cosine of the angle between those sides
minus 2 times the productof those other sides
One side squared
sum of each of the other sides squared
Now punch buttons on your calculator to find a.
It will be square root of right hand side.
a 2.9930
CAUTION Don't forget order of operations
powers then multiplication BEFORE addition and
subtraction
12
We'll label side a with the value we found.
We now have all of the sides but how can we find
an angle?
B
3
19.21
2.993
80
C
80.79
Hint We have an angle and a side opposite it.
1
C is easy to find since the sum of the angles is
a triangle is 180
When taking arcsin, use 2nd answer on your
calculator for accuracy!
13
Ex. 2 Solve a triangle where a 5, b 8 and c
9
Draw a picture.
B
9
This is SSS
5
Do we know an angle and side opposite it? No, so
we must use Law of Cosines.
C
A
84.26
8
Let's choose to find angle C first.
times the cosine of the angle between those sides
minus 2 times the productof those other sides
One side squared
sum of each of the other sides squared
CAUTION Don't forget order of operations
powers then multiplication BEFORE addition and
subtraction
14
How can we find one of the remaining angles?
Do we know an angle and side opposite it?
B
9
62.18
5
?
84.26
33.56
A
8
Yes, so use Law of Sines.
15
Too easy, whats the catch?

• After we use L.O.C. we need to use law of sines
  to find the remaining sides and angles.
• The range of arcsin is -90 deg to 90 deg, but
  what if the angle is obtuse? Then taking the
  arcsin wont get us the correct angle!
• To avoid this problem When using L.O.S. after
  L.O.C. always find the smallest angle FIRST The
  smallest angle has to be acute since there cant
  be more than one obtuse angle in a triangle.
• Then use the triangle sum thm to find the 3rd
  angle.

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Try it on your own! 1

• Find the three angles of the triangle ABC if

C
8
6
A B
12
17
Try it on your own! 2

• Find the remaining angles and side of the
  triangle ABC if

C
16
80
A B
12
18
Summary What could we use to solve the
following triangles?
70
30
80
Uh, nothing. Its AAA
19
Summary What could we use to solve the
following triangles?
Do we know an angle and side opposite it?
Could we find it?
16
20
80
ASA although we dont know an angle and side
opposite each other we can find the 3rd angle
then do law of sines
20
Summary What could we use to solve the
following triangles?
Do we know an angle and side opposite it?
16
20
80
AAS law of sines
21
Summary What could we use to solve the
following triangles?
Do we know an angle and side opposite it?
16
20
80
ASS, we can use law of sines but need to check
for 1, 2 or no triangles.
22
Summary What could we use to solve the
following triangles?
Do we know an angle and side opposite it?
16
80
12
SAS dont know (and cant find) angle and side
opposite Law of Cosines
23
Summary What could we use to solve the
following triangles?
Do we know an angle and side opposite it?
16
20
12
SSS dont know (and cant find) angle and side
opposite Law of Cosines
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Wing Span
C

• The leading edge ofeach wing of theB-2 Stealth
  Bombermeasures 105.6 feetin length. The angle
  between the wing's leading edges is 109.05.
  What is the wing span (the distance from A to C)?
• Note these are the actual dimensions!

A
25
Wing Span
C
A
26
Navigational Bearings

• The direction to a point is stated as the number
  of degrees east or west of north or south. For
  example, the direction of
• A from O is N30ºE.B is N60ºW from O.C is S70ºE
  from O.D is S80ºW from O

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H Dub

• 6-2 Pg. 443 2-16even, 17-22all, 29, 34, 35

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Practice 1
Practice 2
29
(No Transcript)
30
(No Transcript)
31
Do you see a pattern?
Use these to findmissing sides
Use these to find missing angles
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Practice 1
33
Practice 2