Answer:
Step-by-step explanation:
Since we have one angle with an opposite side and another side known, we are dealing with an ambiguous case. There are either
0
,
1
or
2
solutions to this triangle.
We start by determining the measure of
A
using the law of Sines, since we already know the measure of side
a
.
sin
A
a
=
sin
B
b
sin
A
14
=
sin
105
˚
23
A
=
arcsin
(
14
sin
105
˚
23
)
A
=
36
˚
There is only one solution to this triangle, because if it was, then the alternative measure of A would be
180
˚
−
36
˚
=
144
˚
, which when added to
B
, would make the sum of the angles in the triangle exceed
180
˚
.
We can now use the measure of angles
A
and
B
to solve for angle C.
A
+
B
+
C
=
180
˚
36
˚
+
105
˚
+
C
=
180
˚
C
=
39
˚
The last step is using this information to reapply the law of sines and determine the length of side
c
.
sin
B
b
=
sin
C
c
sin
105
˚
23
=
sin
39
˚
c
c
=
23
sin
39
˚
sin
105
˚
c
=
15
In summary
The triangle has the following measures.
∙
A
=
36
˚
∙
a
=
14
units
∙
B
=
105
˚
∙
b
=
23
units
∙
C
=
39
˚
∙
c
=
15
units
Hopefully this helps!
