Given Triange ABC with

an area of 100.2 a = 32.5 c = 29.2
Find angle B, expressed in degrees.

Area = (1/2)ac*sinB = 100.2,

(1/2)*32.5*29.2sinB = 100.2,
474.5sinB = 100.2,
sinB = 0.2112,
B = 12.2 Deg.

To find angle B in triangle ABC, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Since the area of triangle ABC is given as 100.2, we can write:

100.2 = (1/2) * a * h

Since the lengths of sides a and c are given as 32.5 and 29.2 respectively, we can substitute them into the equation:

100.2 = (1/2) * 32.5 * h

Now, let's solve for the height (h):

100.2 = 16.25 * h
h = 100.2 / 16.25
h ≈ 6.16

Now, we can use the Law of Sines to find angle B:

sin(B) / c = sin(A) / a

Substituting the values we have:

sin(B) / 29.2 = sin(A) / 32.5

Now, let's find sin(A):

sin(A) = h / c
sin(A) = 6.16 / 29.2
sin(A) ≈ 0.211

Now, let's plug in the values into the Law of Sines equation:

sin(B) / 29.2 = 0.211 / 32.5

To isolate sin(B), we multiply both sides by 29.2:

sin(B) ≈ (0.211 / 32.5) * 29.2
sin(B) ≈ 0.189

Finally, we can find angle B by taking the inverse sine (sin⁻¹) of 0.189:

B ≈ sin⁻¹(0.189)
B ≈ 10.9 degrees

Therefore, angle B is approximately 10.9 degrees.

To find angle B in Triangle ABC, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, we know side a = 32.5, side c = 29.2, and the area of the triangle is 100.2. To find angle B, we need to find side b, and then we can use the law of cosines to calculate angle B.

First, let's find side b.

The formula for the area of a triangle is given by the formula:

Area = (1/2) * base * height

In triangle ABC, side b is the base, and the height is the perpendicular distance from angle B to side b. Let's represent the height as h.

Area = (1/2) * b * h

Given that the area is 100.2, we can rearrange the formula to solve for h:

h = (2 * Area) / b

Substituting the given values:

h = (2 * 100.2) / b
h = 200.4 / b

Now, we can use the formula for the area of a triangle again to solve for side b:

Area = (1/2) * base * height
100.2 = (1/2) * b * h
100.2 = (1/2) * b * (200.4 / b)

Simplifying the equation gives:

100.2 = 100.2

This means that the value of b does not affect the area of the triangle. Therefore, any value of side b will give us the same area of 100.2. It is not possible to uniquely determine the value of b given this information.

Since we cannot determine side b, we also cannot calculate angle B using the law of cosines. Therefore, without further information, we cannot find angle B in Triangle ABC.