2
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?
1512.5
26°No side opposite from any angle to
get the ratio
No side opposite from any angle to
get the ratio
Let's consider types of triangles with the three pieces of information shown below.
SAS
You may have a side, an angle, and then another side
AAA
You may have all three angles.
SSS
You may have all three sides
This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down"
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.
AAA
LAW OF COSINES
Cabbac cos2222
Baccab cos2222
Abccba cos2222 LAW OF COSINES
ab
cbaC
2cos
222
ac
bcaB
2cos
222
bc
acbA
2cos
222
Use these to findmissing sides
Use these to find missing angles
Do you see a pattern?
5
Deriving the Law of Cosines
• Write an equationusing Pythagorean theorem for shaded triangle.
b h a
k c - kA B
C
c
sin
cos
h b A
k b A
2 22
2 2 2 2 2 2
2 2 2 2 2
2 2 2
sin cos
sin 2 cos cos
sin cos 2 cos
2 cos
a b A c b A
a b A c c b A b A
a b A A c c b A
a b c c b A
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).
Solve a triangle where b = 1, c = 3 and A = 80°
Draw a picture.
80
B
C
a
1
3
Do we know an angle and side opposite it? No so we must use Law of Cosines.
Hint: we will be solving for the side opposite the angle we know.
This is SAS
Abccba cos2222 times the cosine of the angle between
those sides
One side squared
2a
sum of each of the other sides
squared
minus 2 times the productof those
other sides
312 80cos22 31
Now punch buttons on your calculator to find a. It will be square root of right hand side.
a = 2.99
CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction
We'll label side a with the value we found.
We now have all of the sides but how can we find an angle?
80
B
C
2.99
1
3
Hint: We have an angle and a side opposite it.
80.77
B is easy to find since the sum of the angles is a triangle is 180°
19.23
1
sin
99.2
80sin B
77.8099.2
80sin3B
23.197.8080180
85.1815.81 andor If you found C first
Cabbac cos2222
Solve a triangle where a = 5, b = 8 and c = 9
Draw a picture. B
C
5
8
9
Do we know an angle and side opposite it? No, so we must use Law of Cosines.
Let's use largest side to find largest angle first.
This is SSS
times the cosine of the angle between
those sides
One side squared
29
sum of each of the other sides
squared
minus 2 times the productof those
other sides
852 Ccos22 85 CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction
A
Ccos808981
80
8cos
C26.84
10
1cos 1
C
84.26
How can we find one of the remaining angles?
B
5
8
9Do we know an angle and side opposite it?
A 84.26
62.18
33.56
Yes, so use Law of Sines.
56.3318.6226.84180 A
8
sin
9
26.84sin B
Bsin9
26.84sin8 18.62
9
26.84sin8sin 1
11
Try it on your own! #1
• Find the three angles of the triangle ABC if
86
A B
C
28.117,34.36,38.26 CBA
12,8,6 cba
12
12
Try it on your own! #2
• Find the remaining angles and side of the triangle ABC if
16
80A B
C
33.40,67.59,26.18 CBa
80,12,16 Amcb
12
13
Wing Span
• 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
C
14
Wing Span
A
C
Baccab cos2222
05.109cos)6.105)(6.105(26.1056.105 222 b
46.727972.223022 b
.172 ftb