The%20Laws%20of%20SINES

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The Laws of SINES
and COSINES
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The Law of SINES
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 SINES when given ...

• AAS
• ASA
• SSA (the ambiguous case)

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Example 1

• You are given a triangle, ABC, with angle A
  70, angle B 80 and side a 12 cm. Find the
  measures of angle C and sides b and c.
• In this section, angles are named with capital
  letters and the side opposite an angle is named
  with the same lower case letter.

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Example 1 (cont)
The angles in a ? total 180, so angle C
30. Set up the Law of Sines to find side b
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Example 1 (cont)
Set up the Law of Sines to find side c
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Example 1 (solution)
Angle C 30 Side b 12.6 cm Side c 6.4 cm
Note We used the given values of A and a in both
calculations since your answer is more accurate
if you do not use rounded values in calculations.
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Example 2

• You are given a triangle ABC, with angle C
  115, angle B 30 and side a 30 cm. Find the
  measures of angle A and sides b and c.

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Example 2 (cont)
To solve for the missing sides/angles, we must
have an angle/side opposite pair to set up the
first equation. We MUST find angle A because the
only side given is side a. The angles in a ?
total 180, so angle A 35.
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Example 2 (cont)
Set up the Law of Sines to find side b
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Example 2 (cont)
Set up the Law of Sines to find side c
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Example 2 (solution)
Angle A 35 Side b 26.2 cm Side c 47.4 cm
Note Use the Law of Sines whenever you are
given 2 angles and one side!
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The Ambiguous Case (SSA)

• When given SSA (two sides and an angle that is
  NOT the included angle) , the situation is
  ambiguous. The dimensions may not form a
  triangle, or there may be 1 or 2 triangles with
  those dimensions. We first go through a series
  of tests to determine how many (if any) solutions
  exist.

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The Ambiguous Case (SSA)
In the following examples, the given angle will
always be angle A and the given sides will be
sides a and b. If you are given a different set
of variables, feel free to change them to
simulate the steps provided here.
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The Ambiguous Case (SSA)
Situation I Angle A is obtuse If angle A is
obtuse there are TWO possibilities.
If a b, then a is too short to reach side c - a
triangle with these dimensions is impossible.
If a gt b, then there is ONE triangle with these
dimensions.
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The Ambiguous Case (SSA)
Situation I Angle A is obtuse - EXAMPLE
Given a triangle with angle A 120, side a 22
cm and side b 15 cm, find the other
dimensions.
Since a gt b, these dimensions are possible. To
find the missing dimensions, use the Law of Sines
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The Ambiguous Case (SSA)
Situation I Angle A is obtuse - EXAMPLE
Angle C 180 - 120 - 36.2 23.8 Use Law of
Sines to find side c
Solution angle B 36.2, angle C 23.8,
side c 10.3
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The Ambiguous Case (SSA)
Situation II Angle A is acute If angle A is
acute there are SEVERAL possibilities.
Side a may or may not be long enough to reach
side c. We calculate the height of the
altitude from angle C to side c to compare it
with side a.
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The Ambiguous Case (SSA)
Situation II Angle A is acute
First, use SOH-CAH-TOA to find h
Then, compare h to sides a and b . . .
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If a lt h, then NO triangle exists with these
dimensions.
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If h lt a lt b, then TWO triangles exist with these
dimensions.
If we open side a to the outside of h, angle B
is acute.
If we open side a to the inside of h, angle B
is obtuse.
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If h lt b lt a, then ONE triangle exists with these
dimensions.
Since side a is greater than side b, side a
cannot open to the inside of h, it can only open
to the outside, so there is only 1 triangle
possible!
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If h a, then ONE triangle exists with these
dimensions.
If a h, then angle B must be a right angle and
there is only one possible triangle with these
dimensions.
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The Ambiguous Case (SSA)
if angle A is obtuse
if angle A is acute find the height, h
bsinA
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The Law of COSINES
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Use Law of COSINES when given ...

• SAS
• SSS (start with the largest angle!)