1. Unit 5 Day 5
Law of Cosines

2. Happiness begins where selfishness ends. - John Wooden
Warm-up
Happiness begins where selfishness ends. - John Wooden
x = 2.25
x = -5
Solve each triangle using Law of Sines. 3) 4)
C = 40
a = 32.97
b = 13.99
A = 83
a = 10
c = 8.05

3. HW Packet p. 9 odds 7. Two Solutions Case-1 Case-2 43.7 136.3
43.7 
102.3  
148.7 units units
9. One Solution
20.2
34.8
62.7 units
Questions 11 & 13 Require Law of Cosines to solve (SAS).

4. HW Packet p. 9 evens 6. One Solution 54 63  44 units 8. One Solution
23
129
248.3 units
10. Two Solutions
Case-1 Case-2
53.8 
95.2  
49.4 units units
12.
Case-1 Case-2
17.5 
146.5  1.5 
22.03 units 1.04 units
Questions 11 & 13 Require Law of Cosines to solve (SAS).

5. Law of Cosines

7. a2 = b2 + c2 – 2bcCosA a2 = 102 + 82 – 2(10)(8)Cos(60)
Ex. 1 Use Law of Cosines to find a.
Notice this arrangement is SAS.
a2 = b2 + c2 – 2bcCosA
a2 = – 2(10)(8)Cos(60)
a2 = 164 – 160Cos(60)

8. r2 = s2 + q2 – 2sqCosR 232 = 372 + 182 – 2(37)(18)Cos(R)
Ex. 2 Use Law of Cosines to find mR.
Notice this arrangement is SSS.
r2 = s2 + q2 – 2sqCosR
232 = – 2(37)(18)Cos(R)
529 = 1693 – 1332Cos(R)
R =cos-1(-1164/1332)
R = 29.1

9. Ex. 3
Tips for steps: You’ll need 3 steps just as you did with Law of Sines Problems
Think BIG – use law of Cosines with CAPITAL letter (given angle) first
Short Side Second with Law of Sines
Subtract angles from 180°
k2 = – 2(14)(18)Cos(51)
Twalden: These Steps are critical for Law of Cosines IF you are doing Law of Sines for step 2. Chelsea Mauldin tells me that if the students use Law of Cosines for all the steps, the problems work fine too.
Sin-1
L = 180 – 51 – = 79.19

10. Extra Example – not in notes - Solving given SSS
Solve triangle ABC if a=8, b=10, and c=5.
Remember, if no picture is given, draw one! 
Tips for steps: You’ll need 3 steps just as you did with Law of Sines Problems
Think BIG – use law of Cosines with BIGGEST side first
Short Side Second with Law of Sines
Subtract angles from 180°
102 = – 2(5)(8)Cos(B)
Twalden: The essential step here is Step 1..big first. Short Side Second is not essential here BUT it is for the other case…so I usually get my AFM students to do short side 2nd for both so they don’t have to remember which it’s essential for. Chelsea Mauldin tells me that if the students use Law of Cosines for all the steps, the problems work fine too.
Sin-1
L = 180 – 97.9 – = 52.43

11. m<A= 34o , m<B= 108o , m<C= 38o
Ex. 4, 5 YOU TRY
Solve each triangle.
a = 33 , m<B= 31o , m<C= 58
m<A= 34o , m<B= 108o , m<C= 38o

12. What type of triangle do you have? Ambiguous Case Law of Sines
Which Formula Do I Use?
What type of triangle do you have?
RIGHT
OBLIQUE
Pythagorean Theorem
& Trig Ratios
AAS,ASA
SSS,SAS
SSA
Ambiguous Case Law of Sines
Law of Sines
Law of Cosines

13. Practice!

14. Capital letters are angles, lowercase letters are sides
Practice Answers!
Capital letters are angles, lowercase letters are sides
B = 86
C = 56
D = 38
H = 31
G = 109
g = 14.7
p = 6.9
M = 79
Q = 63
22.) B = 61, b = 5.4, a = 4.1
23.) c = 6.3, A = 80, B = 63
24.) A = 30, B = 39, C = 111
25.) B = 99, b = 31.3, a = 25.3