Question

1. find ( mangle b ) to the nearest degree.

(image of triangle abc with: side ac (b) = 4 cm, angle at c = ( 55^circ ), side ab (c) = 4.5 cm. multiple choice options: ( 47^circ ), ( 66^circ ), ( 78^circ ), ( 67^circ ))

Explanation:

Step1: Identify triangle parts

We have triangle \( ABC \) with \( c = 4.5 \, \text{cm} \), \( b = 4 \, \text{cm} \), \( \angle C = 55^\circ \). We can use the Law of Sines: \( \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \), but first, let's check if we can find \( \angle A \) or use Law of Sines for \( \angle B \). Wait, actually, let's list the sides: side \( b \) is opposite \( \angle B \)? No, wait: in triangle notation, side \( a \) is opposite \( \angle A \), side \( b \) opposite \( \angle B \), side \( c \) opposite \( \angle C \). Wait, no: standard notation is side \( a \) opposite \( \angle A \), side \( b \) opposite \( \angle B \), side \( c \) opposite \( \angle C \). Wait, in the diagram, side \( AC = b = 4 \, \text{cm} \) (opposite \( \angle B \)), side \( AB = c = 4.5 \, \text{cm} \) (opposite \( \angle C \)), and \( \angle C = 55^\circ \). So using Law of Sines: \( \frac{\sin B}{b}=\frac{\sin C}{c} \).

Step2: Apply Law of Sines

Substitute \( b = 4 \), \( c = 4.5 \), \( \angle C = 55^\circ \):
\( \sin B=\frac{b \cdot \sin C}{c}=\frac{4 \cdot \sin 55^\circ}{4.5} \)
Calculate \( \sin 55^\circ \approx 0.8192 \):
\( \sin B=\frac{4 \cdot 0.8192}{4.5}=\frac{3.2768}{4.5}\approx 0.7282 \)

Step3: Find \( \angle B \)

Take the inverse sine: \( \angle B=\arcsin(0.7282)\approx 46.7^\circ \approx 47^\circ \) (to nearest degree).

Answer:

\( 47^\circ \) (corresponding to the option with \( 47^\circ \))