Question

determine the remaining sides and angles of the triangle abc.

the measure of angle b is 43.1 °
(simplify your answer. type an integer or a decimal.)
the length of side a is 36.1 ft.
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)
the length of side b is ft.
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

Explanation:

Step1: Recall Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where $a, b, c$ are the sides opposite angles $A, B, C$ respectively. We know angle $A = 34.7^\circ$, angle $B=43.1^\circ$, angle $C = 102.2^\circ$, and side $c = 61.9$ ft (side $c$ is opposite angle $C$, side $a$ is opposite angle $A$, side $b$ is opposite angle $B$).

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

We use the ratio $\frac{b}{\sin B}=\frac{c}{\sin C}$. Rearranging for $b$, we get $b=\frac{c\times\sin B}{\sin C}$.
Substitute $c = 61.9$ ft, $\sin B=\sin(43.1^\circ)$, $\sin C=\sin(102.2^\circ)$.
First, calculate $\sin(43.1^\circ)\approx0.684$, $\sin(102.2^\circ)\approx0.978$.
Then, $b=\frac{61.9\times0.684}{0.978}$.
Calculate numerator: $61.9\times0.684 = 61.9\times0.6 + 61.9\times0.084=37.14+5.20=42.34$.
Then, $b=\frac{42.34}{0.978}\approx43.3$.

Answer:

43.3