Problem 4) 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.

Answer 1

To solve this problem, we can use the Law of Sines, which states that the ratio of each side length to the sine of its opposite angle is constant. In this case, we can write:

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

We are given that angle B is pi/3 and side b is 8 feet. We want to find the lengths of sides a and c.

Let's begin by finding angle A. Since the sum of the angles in a triangle is pi radians (or 180 degrees), we can find angle A using the formula:

angle A = pi - angle B - angle C

Substituting in the given value for angle B, we have:

angle A = pi - (pi / 3) - angle C

Simplifying, we have:

angle A = 2pi / 3 - angle C

Now, let's use the Law of Sines to find side a. We can write:

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

Substituting in the known values, we have:

sin(2pi / 3 - angle C) / a = sin(pi / 3) / 8

Since sin(2pi / 3 - angle C) is equal to sin(angle C) (from the periodicity of sine), we can simplify to:

sin(angle C) / a = sin(pi / 3) / 8

Cross multiplying, we have:

8sin(angle C) = asin(pi / 3)

Dividing both sides by sin(angle C), we have:

8 = a / sin(angle C) * sin(pi / 3)

Since sin(pi / 3) is equal to sqrt(3) / 2, we can simplify to:

8 = a / sin(angle C) * sqrt(3) / 2

Multiplying both sides by 2 to isolate a, we have:

16 = a / sin(angle C) * sqrt(3)

Now, let's solve for side c. We can use the same formula as before:

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

Substituting in the known values, we have:

sin(2pi / 3 - angle C) / a = sin(pi / 3) / 8 = sin(angle C) / c

Since sin(2pi / 3 - angle C) is equal to sin(angle C), we can simplify to:

sin(angle C) / a = sin(pi / 3) / 8 = sin(angle C) / c

Cross multiplying, we have:

sin(angle C) * 8 = asin(pi / 3)

Dividing both sides by sin(angle C), we have:

8 = a / sin(angle C) * sin(pi / 3)

Since sin(pi / 3) is equal to sqrt(3) / 2, we can simplify to:

8 = a / sin(angle C) * sqrt(3) / 2

Multiplying both sides by 2 to isolate c, we have:

16 = a / sin(angle C) * sqrt(3)

Therefore, the lengths of sides a and c are both equal to 16 / (sin(angle C) * sqrt(3)) feet.

Answer 2

write in only numbers for the steps

Answer 3

Given angle B = π/3 and side b = 8 feet.

1) Find angle A:
angle A = π - π/3 - angle C = (2π/3 - angle C)

2) Use the Law of Sines to find side a:
8 = a/sin(angle C) * √3/2
a = 8sin(angle C) * 2/√3

3) Use the Law of Sines to find side c:
8 = c/sin(angle C) * √3/2
c = 8sin(angle C) * 2/√3

Therefore, the lengths of sides a and c are both equal to 16/(sin(angle C) * √3) feet.