Find the solutions for a triangle with a = 11.4, b = 13.7, and c = 12.2

Solution:

Given, the side lengths of a triangle are a = 11.4, b = 13.7 and c = 12.2

We have to find the solutions of the given triangle.

Since the three sides are known, the angles of the triangle can be found out using the law of cosine.

By using Law of Cosine,

Cos A = (b² + c² - a²) / 2bc

Cos A = [(13.7)² + (12.2)² - (11.4)²] / 2(13.7)(12.2)

= (187.69 + 148.84 - 129.96) / 334.28

= 206.57/334.28

= 0.6180

Now, A = cos⁻¹(0.6180)

Therefore, ∠A = 51.83°

Cos B = a² + c² - b²/2ac

Cos B = (11.4)² + (12.2)² - (13.7)² / 2(11.4)(12.2)

= (129.96 + 148.84 - 187.69) / 278.16

= 91.11/278.16

= 0.3275

Now, B = cos⁻¹(0.3275)

Therefore, ∠B = 70.88°

Cos C = a² + b² - c²/2ab

Cos C = (11.4)² + (13.7)² - (12.2)² / 2(11.4)(13.7)

= (129.96 + 187.69 - 148.84) / 312.36

= 168.81/312.36

= 0.5404

Now, C = cos⁻¹(0.5404)

Therefore, ∠C = 57.29°

---

Find the solutions for a triangle with a = 11.4, b = 13.7, and c = 12.2

Summary:

The solutions for a triangle with a =11.4, b =13.7, and c =12.2 are ∠A = 51.83°, ∠B = 70.88° and ∠C = 57.29°.