The Cosine Rule.

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The Cosine Rule.
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Apply Pythagoras to triangle CDB.
a2
h2

(c - x) 2
Square out the bracket.
a2
h2

c2
-2cx
x2
b2
a2
b2

c2
-2cx
What does h2 and x2 make?
a2
b2

c2
-2cbcosAo
What does the cosine of Ao equal?
x
We now have
cos Ao
Make x the subject
b
Substitute into the formula
x
bcosAo
The Cosine Rule.
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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
Note the pattern of sides and angle.
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Using The Cosine Rule.
Example 1.
Find the unknown side in the triangle below
Identify sides a,b,c and angle Ao
Write down the Cosine Rule.
c
12
Ao
43o
a
L
b
5
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 a.
a 9.02m
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Example 2.
Find the length of side M.
Identify the sides and angle.
a M
b 12.2
C 17.5
Ao 137o
Write down Cosine Rule and substitute values.
a2 12.22 17.52 ( 2 x 12.2 x 17.5 x cos 137o
)
a2 148.84 306.25 ( 427 x 0.731 )
Notice the two negative signs.
a2 455.09 312.137
a2 767.227
a 27.7m
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What Goes In The Box ? 1.
Find the length of the unknown side in the
triangles below
G 12.4cm
L 47.5cm
M 5.05m
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Finding Angles Using The Cosine Rule.
Consider the Cosine Rule again
We are going to change the subject of the formula
to cos Ao
Turn the formula around
b2 c2 2bc cos Ao a2
Take b2 and c2 across.
-2bc cos Ao a2 b2 c2
Divide by 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.
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Finding An Angle.
Example 1
Use the formula for Cos Ao to calculate the
unknown angle xo below
Ao xo
a 11
b 9
c 16
Write down the formula for cos Ao
Identify Ao and a , b and c.
Substitute values into the formula.
Cos Ao
0.75
Calculate cos Ao .
Ao 41.4o
Use cos-1 0.75 to find Ao
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Example 2.
Find the unknown angle in the triangle below
Write down the formula.
Identify the sides and angle.
Substitute into the formula.
Ao yo
a 26
b 15
c 13
Find the value of cosAo
The negative tells you the angle is obtuse.
cosAo
- 0.723
Ao
136.3o
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What Goes In The Box ? 2
Calculate the unknown angles in the triangles
below
ao 111.8o
bo
bo 37.3o
co 128.2o