The Law of Sines

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Chapter 14.1

• The Law of Sines

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We know that Trigonometry can help us solve
right triangles. But not all triangles are right
triangles. Fortunately, Trigonometry can help us
solve non-right triangles as well. Non-right
triangles are know as oblique triangles. There
are two categories of oblique trianglesacute and
obtuse.
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Acute Triangles
In an acute triangle, each of the angles is less
than 90º.
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Obtuse Triangles
In an obtuse triangle, one of the angles is
obtuse (between 90º and 180º). Can there be two
obtuse angles in a triangle?
Of course not!
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Law of Sines

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The Law of Sines is used when we know any two
angles and one side or when we know two sides and
an angle opposite one of those sides.
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There are two cases where one side and two angles
are known ASA or SAA
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Disclaimer Do not assume that any
triangles are drawn to scale.
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Lets do an example involving ASA. From the
model, we need to determine a, b, and ? using
the law of sines.
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First off, 42º 61º ? 180º so that ?
77º. (Knowledge of two angles yields the third!)
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Now by the law of sines, we have the following
relationships
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Lets solve for our unknowns
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Now, an example involving SAA From the model, we
need to determine a, b, and ? using the law of
sines. Note ? 110º 40º 180º so that ?
30º
b
a
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By the law of sines, we have the following
relationships
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Therefore,
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The Ambiguous Case SSA In this case, you may
have information that results in one triangle,
two triangles, or no triangles.
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Example 1 of SSA Two sides and an angle
opposite one of the sides are given. Lets try
to solve this triangle.
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By the law of sines,
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Thus,
Therefore, there is no value for ? that exists!
No triangle is possible!
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Example 2 of SSA
Two sides and an angle opposite one of the sides
are given. Lets try to solve this triangle.
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By the law of sines,
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So that,
Interesting! Lets see if one or both of these
angle measures makes sense.
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Case 1 Case 2
Both triangles are valid! Therefore, we have two
possible cases to solve.
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Finish Case 1
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Finish Case 2
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Wrapping it up, here are our two solutions
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Example 3 of SSA
Two sides and an angle opposite one of the sides
are given. Lets try to solve this triangle.
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By the law of sines,
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(No Transcript)
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Note Only one is legitimate!
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Thus, we have only one triangle.
Now lets find b.
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By the law of sines,
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Finally, we have
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Example Finding the Height of a Telephone Pole

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The Area of a TriangleUsing Trigonometry
We can find the area of a triangle if we are
given any two sides of a triangle and the measure
of the included angle. (SAS)
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Example Find the area of given a 32 m, b 9
m, and
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Homework pg. 891 893 11 47 e.o.o. and
46, 50, 53, 55

• Now go and practice. Remember, practice makes
  perfect!