Find all solutions to the following triangle. (Round your answers to one decimal place. If either triangle is not possible, enter NONE in each corresponding answer blank.)
A = 115.2¡ã, a = 43.6 cm, b = 23.1 cm
First triangle (assume B ¡Ü 90¡ã):
B
=
¡ã
C
=
¡ã
c
=
cm
Second triangle (assume B' > 90¡ã):
B'
=
¡ã
C'
=
¡ã
c'
=
cm
I am not familiar with your symbols.
sinB/b = sinA/a.
sinB/23.1 = sin115.2/43.6
Multiply both sides by 23.1
sinB = 23.1*sin115.2/43.6 = 0.47939
B = 28.6o.
CosA = b^2+c^2-a^2/2bc.
Cos115.2 = (23.1)^2+c^2-(43.6)^2/46.2c
-0.42578 = (533.61+c^2-1901)/46.2
-0.42578 = (-1367.39+c^2)/46.2c
-0.42578c = (-1367.39+c^2)/46.2
-19.67c = -1367.39+c^2
-c^2-19.67c = -1367.39
c^2+19.67c-1367.39 = 0.
c = 28.43 cm.
C = 180-115.2 -28.6 = 36.2o
To find all the solutions to the given triangle, we will use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
Let's start with the first triangle (assume B ≤ 90°):
1. First, we'll find angle B using the Law of Sines:
sin(B) / b = sin(A) / a
sin(B) = (b * sin(A)) / a
sin(B) = (23.1 cm * sin(115.2°)) / 43.6 cm
sin(B) ≈ 0.5829
B ≈ arcsin(0.5829)
B ≈ 35.9°
2. Now, we can find angle C using the fact that the sum of angles in a triangle is 180°:
C = 180° - A - B
C = 180° - 115.2° - 35.9°
C ≈ 28.9°
3. To find side c, we can use the Law of Sines again:
sin(C) / c = sin(A) / a
c ≈ (sin(C) * a) / sin(A)
c ≈ (sin(28.9°) * 43.6 cm) / sin(115.2°)
c ≈ 15.9 cm
Therefore, the first triangle's dimensions are:
B = 35.9°, C = 28.9°, c = 15.9 cm.
Now let's move on to the second triangle (assume B' > 90°):
1. First, we need to find angle B' using the Law of Sines:
sin(B') / b = sin(A) / a
sin(B') = (b * sin(A)) / a
sin(B') = (23.1 cm * sin(115.2°)) / 43.6 cm
sin(B') ≈ 0.5829
B' = 180° - arcsin(0.5829)
B' ≈ 144.1°
2. To find angle C', we can use the fact that the sum of angles in a triangle is 180°:
C' = 180° - A - B'
C' = 180° - 115.2° - 144.1°
C' ≈ -79.3°
Since a triangle's angles must be positive, the given dimensions do not form a second possible triangle in this case. Therefore, the answer for the second triangle is NONE.
To summarize the solutions to the triangle:
First triangle (assume B ≤ 90°):
B = 35.9°, C = 28.9°, c = 15.9 cm.
Second triangle (assume B' > 90°):
NONE.