Given that angle B is ∏ /3 and side b is 8 feet, determine the lengths of sides an and c. Do not round answers to decimals show step by step

Answer 1

To determine the lengths of sides an and c, we can use the Law of Sines:

The Law of Sines states that for any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant:

a/Sin(A) = b/Sin(B) = c/Sin(C)

Given that angle B is π/3 and side b is 8 feet, we can set up the following equation:

a/Sin(π/3) = 8/Sin(B)

To find the length of side a, we rearrange the equation:

a = (8 * Sin(π/3)) / Sin(B)

Now, we substitute the values and evaluate the expression:

a = (8 * Sin(π/3)) / Sin(π/3)
a = (8 * √3/2) / √3/2
a = (8 * √3) / √3
a = 8 feet

Therefore, the length of side a is 8 feet.

To find the length of side c, we can use the Law of Sines again:

c/Sin(C) = 8/Sin(B)

Since the sum of the angles in a triangle is equal to π radians (180 degrees), we can determine angle C as follows:

C = π - A - B
C = π - π/3 - π/2
C = 5π/6

Now we substitute the values and solve for c:

c/Sin(5π/6) = 8/Sin(π/3)

c = (8 * Sin(5π/6)) / Sin(π/3)

Using the fact that Sin(5π/6) = Sin(π - 5π/6) = Sin(π/6), and Sin(π/3) = √3/2:

c = (8 * Sin(π/6)) / (√3/2)
c = (8 * 1/2) / (√3/2)
c = (4) / (√3/2)
c = (4 * 2) / √3
c = 8/√3

Therefore, the length of side c is (8/√3) feet.