The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the angle opposite that side is the same for all sides and angles. In other words, if a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the angles opposite those sides, then we have:
a/sin(A) = b/sin(B) = c/sin(C)
In the given triangle, we are given the values of B and c, so we can use the Law of Sines to find the other side lengths and angles. We have:
a/sin(A) = c/sin(C)
Since c = 43 and C = 135°, we can plug these values into the equation to find a:
a/sin(A) = 43/sin(135°)
To find the value of a, we just need to solve for a by multiplying both sides of the equation by sin(A). We have:
a = 43 * sin(A) / sin(135°)
We know the value of B, so we can use the Law of Sines again to find the value of A:
sin(A) / a = sin(B) / b
Since B = 8° and b is unknown, we can plug these values into the equation to find a:
sin(A) / a = sin(8°) / b
To find the value of A, we just need to solve for A by dividing both sides of the equation by sin(A)/a. We have:
A = asin(8°) / b
Now that we know the values of A and c, we can use the Law of Sines one more time to find the value of b:
sin(A) / a = sin(B) / b
Since A and a are unknown, we can plug the known values into the equation to find b:
sin(A) / a = sin(B) / b
To find the value of b, we just need to solve for b by multiplying both sides of the equation by sin(B)/b. We have:
b = sin(B) / (sin(A) / a)
At this point, we have all the necessary values to compute the values of a, A, and b. We can plug the known values into the equations we derived above to find the unknown values.
First, let's find the value of a:
a = 43 * sin(A) / sin(135°)
We know the values of A and B, so we can plug these values into the equation to find a:
a = 43 * sin(A) / sin(135°)
= 43 * sin(180° - C - B) / sin(135°)
= 43 * sin(180° - 135° - 8°) / sin(135°)
= 43 * sin(37°) / sin(135°)
To compute the value of sin(37°), we can use a calculator or look up the value in a table of sines.