Given that angle B is ∏ /3 and side b is 8 feet, determine the lengths of sides a and c. Do not round answers to decimals.
To determine the lengths of sides a and c, we will use the Law of Sines.
The Law of Sines states that for any triangle with sides a, b, c and opposite angles A, B, C, the following relationship holds:
a / sin(A) = b / sin(B) = c / sin(C)
In this case, we are given angle B = π/3 and side b = 8 feet. Let's solve for side a first.
Since the sum of the angles in a triangle is always π (180 degrees), angle A = π - B - C.
Since the sum of the angles opposite to their respective sides in a triangle is also always π (180 degrees), angle A is opposite to side a.
So, angle A = π - B - C = π - π / 3 - C = 2π / 3 - C
Using the Law of Sines, we have:
a / sin(A) = b / sin(B)
a / sin(2π / 3 - C) = 8 / sin(π / 3)
Now, we need to determine angle C.
The sum of the angles in a triangle is always π (180 degrees), so angle C = π - A - B = π - 2π / 3 - π / 3 = π / 3
Substituting angle C = π / 3 into the equation:
a / sin(2π / 3 - π / 3) = 8 / sin(π / 3)
a / sin(π / 3) = 8 / sin(π / 3)
a = 8
So, side a is 8 feet.
Now, let's solve for side c.
Using the Law of Sines, we have:
c / sin(C) = b / sin(B)
c / sin(π / 3) = 8 / sin(π / 3)
c = 8
So, side c is also 8 feet.