Question

determine the remaining sides and angles of the triangle abc.
a = 85° (round to the nearest degree as needed.)
b ≈ 8.78 m
(do not round until the final answer. then round to the nearest hundredth as needed.)
c ≈ □ m
(do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 10\space m\), \(A=85^{\circ}\), \(B = 61^{\circ}\), \(C=34^{\circ}\). We want to find \(c\).
So, \(\frac{c}{\sin C}=\frac{a}{\sin A}\).

Step2: Solve for \(c\)

Substitute the values into the formula: \(c=\frac{a\sin C}{\sin A}\).
Since \(a = 10\), \(\sin C=\sin(34^{\circ})\approx0.5592\), \(\sin A=\sin(85^{\circ})\approx0.9962\).
Then \(c=\frac{10\times0.5592}{0.9962}\).
Calculate \(10\times0.5592 = 5.592\).
Then \(c=\frac{5.592}{0.9962}\approx5.61\).

Answer:

\(c\approx5.61\space m\)