Question

determine the remaining sides and angles of the triangle abc. a = 58° (round to the nearest degree as needed.) b ≈ □ m (do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Recall Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ for a triangle \(ABC\) with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\) respectively.
We know \(a = 12\) m, \(A = 58^\circ\), \(B = 65^\circ\), and we need to find \(b\).

Step2: Apply Law of Sines for \(b\)

From $\frac{a}{\sin A}=\frac{b}{\sin B}$, we can solve for \(b\) as \(b=\frac{a\sin B}{\sin A}\).
Substitute \(a = 12\), \(\sin B=\sin65^\circ\), \(\sin A=\sin58^\circ\) into the formula.
First, calculate \(\sin65^\circ\approx0.9063\) and \(\sin58^\circ\approx0.8480\).
Then \(b=\frac{12\times0.9063}{0.8480}\).
Calculate the numerator: \(12\times0.9063 = 10.8756\).
Then divide by \(0.8480\): \(b=\frac{10.8756}{0.8480}\approx12.82\).

Answer:

\(b\approx12.82\)