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# The Cosine Rule.

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The Cosine Rule.
C
b
a
A
B
c
a2 = b2 + c2 -2bccosAo
C
C
b
a
a
h
h
c-x
B
x
A
c-x
D
a2 = h2 + (c - x) 2
c
B
D
Apply Pythagoras to triangle CDB.
a2 = h2 + c2 -2cx + x2 Square out the bracket.
a2 = b2 + c2 -2cx
What does
h2 and
x2 make?
b2
a2 = b2 + c2 -2cbcosAo What does the cosine of Ao equal?
We now have:
a2 = b2 + c2 -2bccosAo
The Cosine Rule.
cos Ao =
x
b
x = bcosAo
Make x the subject:
Substitute into
the formula:
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.
All the triangles below are suitable for use with the Cosine
Rule:
10
6
W
65o
6.2
L
8
147o
M
11
89o
13.8
Note the pattern
of sides and
angle.
Using The Cosine Rule.
Example 1.
Find the unknown side in the triangle below:
L
5m
43o
12m
Identify sides a,b,c and
angle Ao
Write down the Cosine Rule.
o
o
a = L b = 5 c =12 A = 43
a2 = b2 + c2 -2bccosAo
Substitute values and find a2.
a2 = 52 + 122 - 2 x 5 x 12 cos 43o
a2 = 25 + 144 - (120 x 0.731 )
a2 = 81.28
Square root to find вЂњaвЂќ.
a = 9.02m
Example 2.
12.2 m
137o
17.5 m
M
a = M b = 12.2 C = 17.5 Ao = 137o
a2 = b2 + c2 -2bccosAo
Find the length
of side M.
Identify the sides
and angle.
Write down Cosine
Rule and substitute
values.
a2 = 12.22 + 17.52 вЂ“ ( 2 x 12.2 x 17.5 x cos 137o )
a2 = 148.84 + 306.25 вЂ“ ( 427 x вЂ“ 0.731 ) Notice the two
negative signs.
2
a = 455.09 + 312.137
a2 = 767.227
a = 27.7m
What Goes In The Box ? 1.
Find the length of the unknown side in the triangles below:
43cm
(1)
G = 12.4cm
78o
31cm
L
6.3cm
L = 47.5cm
(3)
(2)
M
5.2m
38o
8m
110o
8.7cm
M =5.05m
G
Finding Angles Using The
Cosine Rule.
Consider the Cosine Rule again:
a2 = b2 + c2 -2bccosAo
We are going to change the subject of the formula to cos Ao
b2 + c2 вЂ“ 2bc cos Ao = a2
Turn the formula around:
-2bc cos Ao = a2 вЂ“ b2 вЂ“ c2
Take b2 and c2 across.
a пЂ­b пЂ­c
2
cos A пЂЅ
o
o
2
пЂ­ 2 bc
b пЂ«c пЂ­a
2
cos A пЂЅ
2
2
2 bc
2
Divide by вЂ“ 2 bc.
Divide top and bottom by -1
You now have a formula for
finding an angle if you know
all three sides of the triangle.
Finding An Angle.
Example 1
Use the formula for Cos Ao to calculate the unknown
angle xo below:
b пЂ«c пЂ­a
2
cos A пЂЅ
o
2
2
9cm
11cm
2 bc
xo
Ao = xo
a = 11 b = 9 c = 16
9 пЂ« 16 пЂ­ 11
2
cos A пЂЅ
o
2
2 п‚ґ 9 п‚ґ 16
2
16cm
Write down the formula for cos Ao
Identify Ao and a , b and c.
Cos Ao = 0.75
Substitute values into the formula.
Ao
Calculate cos Ao .
=
41.4o
Use cos-1 0.75 to find Ao
Example 2.
Find the unknown angle in the triangle below:
yo
15cm
13cm
Identify the sides and
angle.
26cm
b пЂ«c пЂ­a
2
cos A пЂЅ
o
2
2
Substitute into the
formula.
2 bc
Ao = yo
a = 26 b = 15
15 пЂ« 13 пЂ­ 26
2
cos A пЂЅ
o
2
2 п‚ґ 15 п‚ґ 13
cosAo = - 0.723
Ao = 136.3o
Write down the formula.
c = 13
2
Find the value of cosAo
The negative tells you
the angle is obtuse.
What Goes In The Box ? 2
Calculate the unknown angles in the triangles below:
(1) 5m
ao
7m
(3)
ao =111.8o 10m
27cm
12.7cm
14cm
co
16cm
(2)
bo
7.9cm
bo = 37.3o
8.3cm
co =128.2o
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