Question

determine the remaining sides and angles of the triangle abc.
a = 58° (round to the nearest degree as needed.)
b ≈ 12.82 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: Recall Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. We know $a = 12$, $A = 58^\circ$, $B = 65^\circ$, $C = 57^\circ$.

Step2: Apply Law of Sines for side c

Using $\frac{a}{\sin A}=\frac{c}{\sin C}$, we can solve for $c$. Rearranging gives $c=\frac{a\sin C}{\sin A}$.
Substitute $a = 12$, $\sin C=\sin 57^\circ$, $\sin A=\sin 58^\circ$.
$\sin 57^\circ\approx0.8387$, $\sin 58^\circ\approx0.8480$.
So $c=\frac{12\times0.8387}{0.8480}=\frac{10.0644}{0.8480}\approx11.87$.

Answer:

$c\approx\boxed{11.87}$