Final answer:
To calculate the two possible sizes of angle A, use the formula for the area of a triangle and the Law of Cosines. Solve the equation to find the two possible values of BC, and substitute them into the Law of Sines to find the two possible sizes of angle A using arcsin.
Step-by-step explanation:
To calculate the two possible sizes of angle A, we can use the formula for the area of a triangle which is 1/2 × base × height.
Given that the area of triangle ABC is 19.6 cm² and AB = 5.9 cm, we can rearrange the formula to solve for height:
Height = 2 × Area / Base = 2 × 19.6 cm² / 5.9 cm = 6.61 cm.
Using the Law of Cosines, we can calculate angle A using the sides AB, AC, and BC:
Cos(A) = (AB² + AC² - BC²) / (2 × AB × AC) = (5.9 cm² + 8.7 cm² - BC²) / (2 × 5.9 cm × 8.7 cm)
To calculate the two possible sizes of angle A, we can solve this equation for BC. Let's solve:
BC = √(AB² + AC² - 2 × AB × AC × Cos(A))
Now we have two possible values for BC. From there, we can substitute these values into the equation for the Law of Sines to find A:
Sin(A) = BC / AB = BC / 5.9 cm
Finally, substitute the values of BC to find the two possible sizes of angle A using arcsin:
A = arcsin(BC₁ / 5.9 cm) and A = arcsin(BC₂ / 5.9 cm).