Answer:
Step-by-step explanation:
a
=
11
,
b
=
11
,
c
=
7
Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.
a
2
=
b
2
+
c
2
−
2
b
c
cos
(
A
)
where
a
=
C
B
,
b
=
A
C
,
c
=
A
B
Solve the equation.
A
=
cos
-1
(
b
2
+
c
2
−
a
2
2
b
c
)
Substitute the known values into the equation.
A
=
cos
-1
(
(
11
)
2
+
(
7
)
2
−
(
11
)
2
2
(
11
)
(
7
)
)
Simplify the results.
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A
=
1.24698531
radians
Convert the angle to degrees.
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A
=
71.44699546
°
The Law of Sines produces an ambiguous angle result. This means that there are
2
angles that will correctly solve the equation. For the first triangle, use the first possible angle value.
Solve for the first triangle.
The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.
sin
(
A
)
a
=
sin
(
B
)
b
=
sin
(
C
)
c
Substitute the known values into the law of sines to find
B
.
sin
(
B
)
11
=
sin
(
71.44699546
)
11
Solve the equation for
B
.
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B
=
71.44699546
,
108.55300453
The sum of all the angles in a triangle is
180
degrees.
71.44699546
+
C
+
71.44699546
=
180
Solve the equation for
C
.
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C
=
37.10600907
For the second triangle, use the second possible angle value.
Solve for the second triangle.
The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.
sin
(
A
)
a
=
sin
(
B
)
b
=
sin
(
C
)
c
Substitute the known values into the law of sines to find
B
.
sin
(
B
)
11
=
sin
(
71.44699546
)
11
Solve the equation for
B
.
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B
=
71.44699546
,
108.55300453
The sum of all the angles in a triangle is
180
degrees.
71.44699546
+
C
+
108.55300453
=
180
Solve the equation for
C
.
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C
=
0
These are the results for all angles and sides for the given triangle.
First Triangle Combination:
A
=
71.44699546
B
=
71.44699546
C
=
37.10600907
a
=
11
b
=
11
c
=
7
Second Triangle Combination:
A
=
71.44699546
B
=
108.55300453
C
=
0
a
=
11
b
=
11
c
=
7
