use law of sines to solve the triangle A=58 degree, a=4.5, b=12.8
sin A/a = sin B/b
.848/4.5 = sin B/12.8
sin B = 2.41
well, that presents a problem.
Try to draw this triangle and you will see that side a must be much longer than 4.5 if b is to be 12.8
To use the Law of Sines to solve a triangle, you need to have one angle and the corresponding side or two angles and one side. In this case, we have angle A (A = 58 degrees) and sides a (a = 4.5) and b (b = 12.8).
To calculate the missing values, we can use the following formula:
a/sinA = b/sinB = c/sinC
1. First, let's find angle B using the Law of Sines. Rearrange the formula to solve for sinB:
b/sinB = a/sinA
sinB = (b * sinA) / a
sinB = (12.8 * sin(58)) / 4.5
sinB ≈ 0.926
Now, use the inverse sine function to find angle B:
B ≈ arcsin(0.926) ≈ 67.2 degrees
2. Now that we have two angles (A = 58 degrees and B ≈ 67.2 degrees), we can find angle C using the fact that the sum of the angles in a triangle is always 180 degrees:
C = 180 - A - B
C = 180 - 58 - 67.2
C ≈ 54.8 degrees
Therefore, the angles of the triangle are A = 58 degrees, B ≈ 67.2 degrees, and C ≈ 54.8 degrees.
3. Finally, we can find the missing side length, side c, using the Law of Sines:
c/sinC = a/sinA
c = (sinC * a) / sinA
c = (sin(54.8) * 4.5) / sin(58)
c ≈ 5.11
Therefore, the missing side length, side c, is approximately 5.11.
To use the Law of Sines to solve a triangle, we need to know either two sides and an opposite angle, or two angles and an opposite side. In this case, we are given angle A = 58 degrees, side a = 4.5, and side b = 12.8.
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. Mathematically, it can be written as:
a / sin(A) = b / sin(B) = c / sin(C),
where a, b, and c are the lengths of the sides, and A, B, and C are the measures of the angles opposite those respective sides.
To solve the triangle, we'll use this formula:
a / sin(A) = b / sin(B),
By substituting the known values:
4.5 / sin(58) = 12.8 / sin(B).
Now, we can solve for sin(B) by cross-multiplying:
sin(B) = (12.8 * sin(58)) / 4.5.
Using a calculator:
sin(B) ≈ 0.9928.
To find angle B, we can take the inverse sine (sin^(-1)) of 0.9928:
B ≈ sin^(-1)(0.9928).
B ≈ 81.8 degrees.
Now, to find angle C, we can use the fact that the sum of the angles in a triangle is always 180 degrees:
C = 180 - A - B.
C = 180 - 58 - 81.8.
C ≈ 40.2 degrees.
Therefore, angle B ≈ 81.8 degrees, and angle C ≈ 40.2 degrees.