Question

use the law of sines. find b to the nearest tenth.
52.9
54.6
26.3
47.7

Explanation:

Step1: Find angle \(C\)

The sum of angles in a triangle is \(180^{\circ}\). So \(C = 180^{\circ}-44^{\circ}-48^{\circ}=88^{\circ}\)

Step2: Apply the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, let \(c = 51\), \(C = 88^{\circ}\), \(A=48^{\circ}\), and we want to find \(b\) (opposite \(A\)). So \(\frac{b}{\sin48^{\circ}}=\frac{51}{\sin88^{\circ}}\)

$$b=\frac{51\times\sin48^{\circ}}{\sin88^{\circ}}$$

Since \(\sin48^{\circ}\approx0.743\), \(\sin88^{\circ}\approx0.999\)

$$b=\frac{51\times0.743}{0.999}\approx\frac{37.893}{0.999}\approx37.93$$

(Wait, no, wrong. Wait, no. Wait, actually, if \(c = 51\) (opposite \(B = 44^{\circ}\)), \(A = 48^{\circ}\), \(B=44^{\circ}\), \(C = 88^{\circ}\). Wait no, original. Wait, in standard notation, side \(a\) is opposite \(A\), side \(b\) opposite \(B\), side \(c\) opposite \(C\). Wait no, in the problem, assume side \(BC\) is \(a\), \(AC\) is \(b\), \(AB\) is \(c\). Wait no, the given side is \(51\) (let's assume \(AC = 51\) is wrong. Wait, no, in the Law of Sines formula \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). If the side given is \(51\) (let's say side \(AC\) is \(51\), opposite \(B = 44^{\circ}\)), and we want to find \(b\) (side \(BC\) opposite \(A = 48^{\circ}\)). Then \(\frac{b}{\sin A}=\frac{51}{\sin B}\)

$$b=\frac{51\times\sin48^{\circ}}{\sin44^{\circ}}$$

Since \(\sin48^{\circ}\approx0.743\), \(\sin44^{\circ}\approx0.695\)

$$b=\frac{51\times0.743}{0.695}=\frac{37.893}{0.695}\approx54.6$$

Answer:

\(54.6\)