Okay, Assume the triangle has the given measurements. Solve for the remaining sides and angles.
A = pi/6; a = 17.4; b = 19.6
sinB/b = sinA/a,
sinB/19.6 = sin30/17.4,
Multiply both sides by 19.6:
sinB = 19.6*sin30/17.4 = 0.5632,
B = 34.3Deg.
C = 180 - (A+B),
C = 180 - (30+34.3) = 115.7deg.
c/sinC = 17.4/sinA,
c/sin115.7 = 17.4/sin30,
Multiply both sides by sin115.7:
c = 17.4*sin115.7/sin30 = 31.4.
To solve for the remaining sides and angles of the triangle, we can use the Law of Sines and Law of Cosines.
Let's start by finding angle B. We can use the Law of Sines, which states that the ratio of the sine of an angle to the length of its opposite side is the same for all angles in a triangle:
sin(A) / a = sin(B) / b
Substituting the given values:
sin(pi/6) / 17.4 = sin(B) / 19.6
To solve for sin(B), we can rearrange the equation:
sin(B) = (sin(pi/6) / 17.4) * 19.6
Now, we can find the value of sin(B):
sin(B) = (1/2) / 17.4 * 19.6
sin(B) = 0.0281
Using the inverse sine (sin^-1) function, we can find the value of angle B:
B = sin^-1(0.0281)
B ≈ 1.61 degrees
Now, let's find side c using the Law of Cosines, which relates the lengths of the sides and the cosine of an angle in a triangle:
c^2 = a^2 + b^2 - 2ab * cos(C)
Substituting the given values:
c^2 = 17.4^2 + 19.6^2 - 2 * 17.4 * 19.6 * cos(pi/6)
c^2 = 303.36 + 384.16 - 2 * 17.4 * 19.6 * cos(pi/6)
c^2 ≈ 163.2
To find the value of c, we take the square root of both sides:
c ≈ sqrt(163.2)
c ≈ 12.77
So, side c is approximately 12.77 units.
Now that we have the values of angles B and side c, we can use the fact that the sum of the angles in a triangle is 180 degrees to find angle C:
C = 180 - A - B
C ≈ 180 - (pi/6) - 1.61
C ≈ 172.95 degrees
Therefore, the remaining measurements of the triangle are:
- Side c ≈ 12.77 units
- Angle B ≈ 1.61 degrees
- Angle C ≈ 172.95 degrees