The sides of the triangle are a=12, b=2, and c=11.88, and the angles of the triangle are ∠A = 88.40°, ∠B = 9.60°, and ∠C = 82°.
How to solve a triangle using the Law of cosines?
Law of cosines:
This law gives the relationship between the lengths of the sides of a triangle w.r.t the cosine of its angle.
I.e., c² = a² + b² - 2ab Cos (C), a² = b² + c² - 2bc Cos (A), and b² = a² + c² - 2ac Cos (B)
Calculation:
The given triangle has C=82°, a = 12 and b = 2.
Using the Law of cosines,
Finding the side length 'c':
c² = a² + b² - 2ab Cos (C)
= (12)² + (2)² - 2(12)(2) Cos (82)
= 141.319
∴ c = 11.88
Finding the angle A:
a² = b² + c² - 2bc Cos (A)
⇒ Cos(A) = [b² + c² - a²]/2bc
⇒ Cos(A) = [2² +(11.88)² - (12)²]/2×2×11.88
⇒ Cos (A) = 0.027
⇒ ∠A = Cos⁻¹(0.027)
∴ ∠A = 88.40°
Finding the angle B:
b² = a² + c² - 2ac Cos (B)
⇒ Cos(B) = [a² + c² - b²]/2ac
⇒ Cos(B) = [(12)² +(11.88)² - (2)²]/2×12×11.88
⇒ Cos (B) = 0.986
⇒ ∠B = Cos⁻¹(0.986)
∴ ∠B = 9.60°
Thus, the three angles of the given triangle are ∠A = 88.40, ∠B = 9.60, and ∠C=82°. The three sides of the given triangle are a=12, b=2, and c=11.8.
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