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
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.