the cosine rule a b c a b c. the cosine rule a b c a b c
TRANSCRIPT
The Cosine Rule
A
B
Ca
b
c
a2 = b
2 + c
2 – 2bc cos A
The Cosine Rule
A
B
Ca
b
c
b2 = a
2 + c
2 – 2cos B
The Cosine Rule
A
B
Ca
b
c
c2 = a
2 + b
2 – 2cos C
When To Use The Cosine Rule.
The Cosine Rule can be used to find a third side of a triangle if you have the other two sides and the angle between them.
6
10
65o
L 89o
13.8
6.2
W
147o
8 11
M
Calculating Sides Using The Cosine
Rule(1).Calculate the length AC
30m
55°
40m
30m
55°
40m
ba
c
b2 = a
2 + c
2 – 2ac cos B
b2 = 30
2+40
2 – 2 30 40 cos 55
b2 = 2500– (2400cos 55)
b2 = 2500– 1376·58
b2 = 2500– 1376·58
b2 = 1123·42
b = 1123·42
b = 33·5 m
Calculating Sides Using The Cosine
Rule(2).
Calculate the length CB
100°13cm5cm
100°13cm5cm
b
a
c
a2 = b
2 + c
2 – 2bc cos A
a2 = 13
2+5
2 – 2 135 cos 100
a2 = 194– (130cos 100)
a2 = 194– (-22·57)
a2 = 216·57
a = 216·57
a = 14·7 cm
a2 = 194– (-22·57)
Rearranging the cosine rule.
a2 = b
2 + c
2 – 2bc cos A
2bc cos A = b2
+c2 – a
2
cosA=b2
+c2–a
2
2bcYou now have a formula for finding an angle if you know all three sides of the triangle.
Calculating Angles Using The Cosine Rule
(1).
Calculate the angle CAB
b
ac
?
cosA=b2
+c2–a
2
2bc
cosA=82
+72–4
2
2 8 7
cosA=82
+72–4
2
2 8 7
cosA=97112
cosA=0·866
A = cos-1 0·866
A = 30°
Calculating Angles Using The Cosine Rule
(2).
Calculate the angle ?
BA
C
cosA=b2
+c2–a
2
2bc
cosA=82
+102–15
2
2 8 10
ba
c
cosA=– 61160
cosA=-0·381
A = cos-1 –0·381
A = 112·4°
cosA=82
+102–15
2
2 8 10