oblique triangles cos and sine law

Question

A metal ring with diameter 1.94 metres is suspended in a horizontal plane by four support ropes each 5.40 m long, which are attached to points equidistant around the ring, and joined together at a swivel above the ring 

 

a) What is the angle between two ropes attached to opposite sides of the ring, at the point that they attach to the swivel, in degrees?

 

b)What is the angle between two adjacent ropes, at the point that they attach to the swivel, in degrees?  

 

Answer 1

To solve this problem, we can use the Law of Cosines. The key is to recognize that the four ropes form triangles with the metal ring, and we need to find the angles where the ropes meet at the swivel.

Step 1: Identify the information given.

We know that:

1. Diameter of the ring = 1.94 metres
2. Radius of the ring = 1.94 / 2 = 0.97 metres
3. Length of each rope = 5.40 metres
4. Four ropes are attached at equally spaced points around the ring.

Because all four ropes have the same length and meet at the same swivel, each pair of ropes forms an isosceles triangle.

An isosceles triangle has two equal sides, which makes the Law of Cosines especially useful.

The Law of Cosines is:

c² = a² + b² - 2ab cos(C)

Where C is the angle between sides a and b, and c is the side opposite that angle.

Part A: Find the angle between two ropes attached to opposite sides of the ring.

Step 2: Determine the three sides of the triangle.

The two ropes are attached to opposite points on the ring, so the distance between their attachment points is the full diameter of the ring.

Therefore:

a = 5.40 m

b = 5.40 m

c = 1.94 m

We want to find angle C, which is the angle between the two ropes at the swivel.

Step 3: Apply the Law of Cosines.

c² = a² + b² - 2ab cos(C)

Substitute:

1.94² = 5.40² + 5.40² - 2(5.40)(5.40)cos(C)

Simplify:

3.7636 = 29.16 + 29.16 - 58.32cos(C)

3.7636 = 58.32 - 58.32cos(C)

Rearrange:

58.32cos(C) = 58.32 - 3.7636

58.32cos(C) = 54.5564

cos(C) = 54.5564 / 58.32

cos(C) ≈ 0.935464

Take the inverse cosine:

C = cos⁻¹(0.935464)

Answer for Part A: Approximately 20.70°

Part B: Find the angle between two adjacent ropes.

Step 4: Determine the distance between adjacent attachment points.

Since the four ropes are equally spaced around the circular ring, the attachment points are separated by 90° around the center.

The distance between two adjacent attachment points is not the diameter. It is the straight-line chord connecting those points.

We can use the Pythagorean theorem because the two radii form a right triangle.

Chord² = 0.97² + 0.97²

Chord² = 0.9409 + 0.9409

Chord² = 1.8818

Chord = √1.8818

Chord ≈ 1.37179 m

Step 5: Apply the Law of Cosines again.

Now we have another isosceles triangle with:

a = 5.40 m

b = 5.40 m

c = 1.37179 m

Using the Law of Cosines:

c² = a² + b² - 2ab cos(C)

Substitute:

1.8818 = 5.40² + 5.40² - 2(5.40)(5.40)cos(C)

1.8818 = 58.32 - 58.32cos(C)

Rearrange:

58.32cos(C) = 58.32 - 1.8818

58.32cos(C) = 56.4382

cos(C) = 56.4382 / 58.32

cos(C) ≈ 0.967733

Take the inverse cosine:

C = cos⁻¹(0.967733)

Answer for Part B: Approximately 14.60°

Final Answers:

A) Angle between opposite ropes = 20.70°

B) Angle between adjacent ropes = 14.60°

Why does this make sense?

The ropes attached to opposite sides of the ring are farther apart at their attachment points, so they form a larger angle at the swivel.

The ropes attached to adjacent points are closer together, so they form a smaller angle.

This is a great example of how the Law of Cosines helps us calculate angles in triangles when we know all three side lengths.