The Law

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The Law
of COSINES
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The Law of COSINES
For any triangle (right, acute or obtuse), you
may use the following formula to solve for
missing sides or angles
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Use Law of COSINES when ...
you have 3 dimensions of a triangle and you need
to find the other 3 dimensions . They cannot be
just ANY 3 dimensions though, or you wont have
enough information to solve the Law of Cosines
equations. Use the Law of Cosines if you are
given

• SAS - 2 sides and their included angle
• SSS

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Example 1 Given SAS
Find all the missing dimensions of triangle ABC,
given that angle B 98, side a 13 and side c
20.
Use the Law of Cosines equation that uses a, c
and B to find side b
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Example 1 Given SAS
Now that we know B and b, we can use the Law of
Sines to find one of the missing angles
Solution b 25.3, C 51.5, A 30.5
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Example 2 Given SAS
Find all the missing dimensions of triangle, ABC,
given that angle A 39, side b 20 and side c
15.
Use the Law of Cosines equation that uses b, c
and A to find side a
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Example 2 Given SAS
Use the Law of Sines to find one of the missing
angles
Important Notice that we used the Law of Sine
equation to find angle C rather than angle B.
The Law of Sine equation will never produce an
obtuse angle. If we had used the Law of Sine
equation to find angle B we would have gotten
87.5, which is not correct, it is the reference
angle for the correct answer, 92.5. If an angle
might be obtuse, never use the Law of Sine
equation to find it.
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Example 3 Given SSS
Find all the missing dimensions of triangle, ABC,
given that side a 30, side b 20 and side c
15.
We can use any of the Law of Cosine equations,
filling in a, b c and solving for one
angle. Once we have an angle, we can either use
another Law of Cosine equation to find another
angle, or use the Law of Sines to find another
angle.
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Example 3 Given SSS
Important The Law of Sines will never produce
an obtuse angle. If an angle might be obtuse,
never use the Law of Sines to find it. For this
reason, we will use the Law of Cosines to find
the largest angle first (in case it happens to be
obtuse).
Angle A is largest because side a is largest
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Example 3 Given SSS
Use Law of Sines to find angle B or C (its safe
because they cannot be obtuse)
Solution A 117.3 B 36.3 C 26.4
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The Law of Cosines
Important The Law of Sines will never produce
an obtuse angle. If an angle might be obtuse,
never use the Law of Sines to find it.
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Additional Resources

• Web Links

• http//oakroadsystems.com/twt/solving.htmSineLaw
• http//oakroadsystems.com/twt/solving.htmDetectiv
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