1. 6.2 The Law of Cosines

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? 15
No side opposite from any angle to get the ratio
26°
12.5

3. 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

4. LAW OF COSINES LAW OF COSINES Do you see a pattern?
Use these to find missing sides
LAW OF COSINES
Use these to find missing angles

5. Deriving the Law of Cosines
Write an equation using Pythagorean theorem for shaded triangle.
b h a
k c - k
A B c

6. 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).

7. a = 2.99 Solve a triangle where b = 1, c = 3 and A = 80° This is SAS B
Draw a picture.
This is SAS B 3 a
Do we know an angle and side opposite it? No so we must use Law of Cosines. 80 C 1
Hint: we will be solving for the side opposite the angle we know.
minus 2 times the product of those other sides
times the cosine of the angle between those 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 = 2.99
CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

8. If you found C first We'll label side a with the value we found. B 3
We now have all of the sides but how can we find an angle? B 3
19.23
2.99 80
Hint: We have an angle and a side opposite it.
80.77 C 1
B is easy to find since the sum of the angles is a triangle is 180°
If you found C first

9. Solve a triangle where a = 5, b = 8 and c = 9
Draw a picture.
This is SSS 9 B 5
Do we know an angle and side opposite it? No, so we must use Law of Cosines. A
84.26 C 8
Let's use largest side to find largest angle first.
minus 2 times the product of those other sides
times the cosine of the angle between those 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

10. How can we find one of the remaining angles?
Do we know an angle and side opposite it? B 9
62.18 5 A
33.56
84.26  8
Yes, so use Law of Sines.

11. Try it on your own! #1 Find the three angles of the triangle ABC if C8 6
A B 12

12. Try it on your own! #2
Find the remaining angles and side of the triangle ABC if C 16 80
A B 12

13. Wing Span C
The leading edge of each wing of the B-2 Stealth Bomber measures feet in length. The angle between the wing's leading edges is °. What is the wing span (the distance from A to C)?
Note these are the actual dimensions! A

14. Wing Span C A

15. H Dub
6-2 Pg. 443 #2-16even, 17-22all, and 29

16. More Practice #1

18. More Practice #2