solve each triangle using either the Law of Sines or the Law of Cosines. If no triangle exists, write “no solution.” Round your answers to the nearest tenth.
A = 23°, B = 55°, b = 9
A = 18°, a = 25, b = 18
In the first case,
C = 180 - 78 = 102 degrees
a = sinA* (b/SinB) = 4.293
c = sinC*(b/sinB) = 10.75
In the second case
sin B = b* (sinA/a) = 0.2225
B = 12.9 or 167.1 degrees
The latter value for B is not possible because the total number of degrees in the triangle would be too high.
C = 149.1 degrees
c = sinC*(a/sinA) = 41.55
To solve each of these triangles, we can use either the Law of Sines or the Law of Cosines. Let's start with the first triangle:
Triangle 1:
Given: A = 23°, B = 55°, b = 9
To use the Law of Sines, we need either two angles and one side or two sides and one angle. In this case, we have one angle (A), one side that is opposite to the unknown angle (b), and one angle (B) that can help us find the remaining angle.
Step 1: Find angle C.
Since the sum of angles in a triangle is 180°, we can find angle C by subtracting the given angles from 180°:
C = 180° - A - B
C = 180° - 23° - 55°
C = 102°
Step 2: Apply the Law of Sines to find the remaining sides.
We can use the Law of Sines with any of the given angles/sides combinations. Let's use angle A and side a.
sin(A) / a = sin(C) / c
sin(23°) / 9 = sin(102°) / c
c = (9 * sin(102°)) / sin(23°)
c ≈ 20.7
So, the sides of the first triangle are approximately:
a ≈ 25, b = 9, c ≈ 20.7
Now let's move to the second triangle:
Triangle 2:
Given: A = 18°, a = 25, b = 18
Since we have one angle (A) and two sides (a, b), we can use the Law of Cosines to find the remaining side.
Step 1: Find angle B.
To find angle B, we can use the Law of Cosines:
cos(B) = (a^2 + c^2 - b^2) / (2ac)
cos(B) = (25^2 + c^2 - 18^2) / (2*25*c)
cos(B) = (625 + c^2 - 324) / (50*c)
cos(B) = (301 + c^2) / (50*c)
We don't know the value of side c yet, so let's use the Law of Sines with angle A and side a to find c.
sin(A) / a = sin(B) / b
sin(18°) / 25 = sin(B) / 18
sin(B) = (18 * sin(18°)) / 25
B = arcsin((18 * sin(18°)) / 25)
B ≈ 32°
Step 2: Find side c.
Using the obtained values of angles A and B, we can find angle C:
C = 180° - A - B
C = 180° - 18° - 32°
C = 130°
Now, apply the Law of Sines to find side c:
sin(C) / c = sin(A) / a
sin(130°) / c = sin(18°) / 25
c = (25 * sin(130°)) / sin(18°)
c ≈ 50.3
So, the sides of the second triangle are approximately:
a = 25, b = 18, c ≈ 50.3
Note: If any side turns out to be negative or c^2 exceeds the sum of the squares of the other two sides, it means no triangle exists with the given values.