Download Law of Sines - Dustin Tench

6.5
The Law of Sines
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The Law of Sines
The figure below describes the possible types of triangles that we will
solve.
(a) ASA or SAA
(b) SSA
(c) SAS
(d) SSS
Case 1 One side and two angles (ASA or AAS)
Case 2 Two sides and the angle opposite one of those
sides (SSA)
Case 3 Two sides and the included angle (SAS)
Case 4 Three sides (SSS)
Cases 1 and 2 are solved by using the Law of Sines; Cases 3 and 4
require the Law of Cosines to initially start solving the triangles.
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The Law of Sines
The Law of Sines says that in any triangle the lengths of
the sides are proportional to the sines of the corresponding
opposite angles.
𝑎
𝑏
𝑐
=
=
Use to find a side.
𝑠𝑖𝑛𝐴
𝑠𝑖𝑛𝐵
𝑠𝑖𝑛𝐶
𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶
Use to find an angle.
=
=
𝑎
𝑏
𝑐
We will follow the convention of
labeling the angles of a triangle as
A, B, and C and the lengths of the
corresponding opposite sides as a,
b, and c.
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Example 1 – Tracking a Satellite (ASA)
A satellite orbiting the earth passes directly overhead at
observation stations in Phoenix and Los Angeles, 340 mi
apart.
At an instant when the satellite is between these two
stations, its angle of elevation is simultaneously observed
to be 60 at Phoenix and 75 at Los Angeles.
How far is the satellite
from Los Angeles?
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Example 2 - AAS
Given m∠A = 26°, m∠C = 56°, and c = 22, solve the
triangle.
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Example 3 - SAS
Given m∠C = 37°, a = 8, b = 11, solve the triangle.
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The Ambiguous Case
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The Ambiguous Case
In Example 1, there was only triangle possible from the
information given.
This is always true for ASA or AAS. But for SSA, there may
be two triangles, one triangle, or no triangle with the given
properties.
For this reason, Case 2 is sometimes called the
ambiguous case.
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The Ambiguous Case
To see why this is so, we show the possibilities when angle
A and sides a and b are given.
• In part (a) no solution is possible, since side a is too
short to complete the triangle.
• In part (b) the solution is a right triangle.
• In part (c) two solutions are possible.
• In part (d) there is a unique triangle with the given
properties.
(a)
(b)
(c)
The ambiguous case
(d)
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Example 4 – SSA, the One-Solution Case
Solve triangle ABC, where A = 45, a = 7 2, and b = 7.
B is acute
B is obtuse
m∠B = _____________
m∠B = _____________
m∠C = _____________
m∠C = _____________
c = ________________
c = ________________
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Example 5 – SSA, the Two-Solution Case
Solve triangle ABC if A = 43.1, a = 186.2, and b = 248.6.
B is acute
B is obtuse
m∠B = _____________
m∠B = _____________
m∠C = _____________
m∠C = _____________
c = ________________
c = ________________
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Example 6 – SSA, the No-Solution Case
Solve triangle ABC, where A = 42, a = 70, and b = 122.
B is acute
B is obtuse
m∠B = _____________
m∠B = _____________
m∠C = _____________
m∠C = _____________
c = ________________
c = ________________
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Example 7 – SSS
Given a = 10, b = 7, and c = 15, solve the triangle.
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WebAssign #14
A tree on a hillside casts a shadow c = 210 ft down the hill.
If the angle of inclination of the hillside is b = 18° to the
horizontal and the angle of elevation of the sun is a = 48°,
find the height of the tree. (Round your answer to the
nearest foot.)
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