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In triangle problems dealing with 2 sides and
2 angles we have seen that the Law of
Sines is used to find the missing item. There are many problems, however,
that deal with all three sides and only one
angle of the triangle. For these problems we have
another method of solution called the Law of
Cosines.
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With the diagram labeled at the
left,
the Law of Cosines is as follows:
Notice that <C and side c are at opposite ends of
the formula. Also, notice the resemblance (in the
beginning of the formula) to the Pythagorean Theorem.
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We can write the Law of Cosines for
each angle around the triangle. Notice in each statement how the
pattern of the letters remains the same.
The Law of Cosines can be used to
find a missing side for a triangle, or a missing angle. Let's
take a look .
Example 1:
In
 , side b
= 12, side c = 20 and m<A = 45º. Find side
a
to the nearest integer.
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Since the only known angle is A, we use the
version of the Law of Cosines dealing with angle A. |
This problem involves all three
sides but only one angle of the triangle. This fits
the profile for the Law of Cosines.
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Example 2:
Find the largest angle, to the nearest tenth of a
degree, of a triangle whose sides are 9, 12 and 18.
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In a triangle, the largest angle is opposite
the largest side. We need to find <B.
Use the Law of Cosines:
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Example 3:
In a parallelogram, the adjacent sides measure 40 cm
and 22 cm. If the larger angle of the parallelogram measure 116º,
find the length of the larger diagonal, to the nearest integer.
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