W11D5 Law of Cos
Warm Up - on white board
1. Solve the triangle
C = 19
c=7
b=5
SSA a >b so ONE solution
sin B = .23
B = 13
A = 148
a = 11.8
Try to find another way B = 167
A+B = 186
2. Solve B = 46
a=6
No Solution
Intro Activity Law of Cos
c=8
a
k
b
x
c−x
Proof for the Law Of Cosines
1.
k 2 + (c − x)2 = b2
k 2 + c2 − 2xc + x2 = b2
k 2 + x2 = a 2
2.
k 2 = a2 − x2
k 2 = b2 − c2 − x2 + 2xc
3.
a2 − x2 = b2 − c2 − x2 + 2xc
a2 = b2 − c2 + 2xc
4.
b2 = a2 + c2 + 2xc
b2 = a2 + c2 − 2ac cos B
5.
x
x = a cos B
a
a2 = b2 + c2 − 2bc cos A
cos B =
b2 = a2 + c2 − 2ac cos B
c2 = b2 + a2 − 2ba cos C
What’s the point... what do we need to solve?
SSS and SAS ok
Lesson 25 Law of Cos
EX 1:
C
6
A
46
8
b2 = a2 + c2 − 2ac cos B
b2 = 36 + 64 − 2(48) cos 46
b2 = 33.3
EX 2:
b = 5.77
A = 48.4
C = 85.6
Solve for the smallest side first!!! To find the biggest angle use the subtraction from 180, not law of sines.
C
7
10
A
B
5
a2 = b2 + c2 − 2bc cos A
52 = 102 + 72 − 2(10)(7) cos A
25 − 100 − 49
cos C =
−140
−124
cos C =
= .88
−140
C = 27.66o
sin A
sin 27.66
=
5
7
sin A = .6499
A = 40.54o
B = 180 − (40.54 + 27.66) = 111.2o
sin B = .93
EX 3:
COS
A = 22
b=6
c=15
a2 = 36 + 225 − 180 cos 22
sin 22
a = 9.7
sin B = 6 13.9
B = 13.3
C=145
EX 4:
a=4
b=6
c=7
sin B = 36−16−19
=
.51
−56
B = 59
A = 35
C = 86
Extra Law of Sines problems
B = 68.2W RON G!
6. B = 40
b = 24
a< b h = 30 sin 40 = 19
sin C = .8
C = 53
C’ = 127
A’ = 13
A = 87
a = 8.4
7. B = 25
b = 12
AAS a<b h = 8.9 sin A = .73
A’ = 133
C’ = 22
a = 21
A = 47
c’ = 10.6
8. A = 35
sin B = .76
B’ = 130
b = 20
C = 95
c’ = 6.7
a = 15
B = 50
C’ = 15
9. A=30
a=7
30
=
1.14
sin C = 16 sin
7
c = 30
c = 16 a < b
No solution
EX 8:
10.
A = 30o
B=B
a= 7
b=b
o
C= Co
c=16
Solution
sin(30)
7
=
sin C
16
No solution
sin C =
16 sin 30
7
= 1.14
a = 37
C = 108
c = 27
c = 26
h=8
a<h