Answer :
To solve the triangle given
=
10
6
∘
A=106
∘
,
=
3
8
∘
C=38
∘
, and
=
12
a=12:
Find angle
B using the triangle sum theorem:
=
18
0
∘
−
−
=
18
0
∘
−
10
6
∘
−
3
8
∘
=
3
6
∘
B=180
∘
−A−C=180
∘
−106
∘
−38
∘
=36
∘
Apply the Law of Sines to find
b:
sin
=
sin
sinA
a
=
sinB
b
Substitute the known values:
12
sin
10
6
∘
=
sin
3
6
∘
sin106
∘
12
=
sin36
∘
b
Calculate
sin
10
6
∘
sin106
∘
(since
sin
10
6
∘
=
sin
(
18
0
∘
−
10
6
∘
)
=
sin
7
4
∘
sin106
∘
=sin(180
∘
−106
∘
)=sin74
∘
):
12
sin
7
4
∘
=
sin
3
6
∘
sin74
∘
12
=
sin36
∘
b
Solve for
b:
=
12
⋅
sin
3
6
∘
sin
7
4
∘
b=
sin74
∘
12⋅sin36
∘
Using a calculator:
sin
3
6
∘
≈
0.5878
,
sin
7
4
∘
≈
0.9613
sin36
∘
≈0.5878,sin74
∘
≈0.9613
≈
12
⋅
0.5878
0.9613
≈
7.36
b≈
0.9613
12⋅0.5878
≈7.36
So,
≈
7.4
b≈7.4.
Find side
c using the Law of Sines:
sin
=
sin
sinA
a
=
sinC
c
Substitute the known values:
12
sin
10
6
∘
=
sin
3
8
∘
sin106
∘
12
=
sin38
∘
c
Calculate
sin
10
6
∘
sin106
∘
:
12
sin
7
4
∘
=
sin
3
8
∘
sin74
∘
12
=
sin38
∘
c
Solve for
c:
=
12
⋅
sin
3
8
∘
sin
7
4
∘
c=
sin74
∘
12⋅sin38
∘
Using a calculator:
sin
3
8
∘
≈
0.6157
sin38
∘
≈0.6157
≈
12
⋅
0.6157
0.9613
≈
7.7
c≈
0.9613
12⋅0.6157
≈7.7
So,
≈
7.7
c≈7.7.
Therefore, the sides of the triangle are approximately
=
12
a=12,
≈
7.4
b≈7.4, and
≈
7.7
c≈7.7.
Answer:
Step-by-step explanation:
