Question

the equation \\(\sin(40^{\circ}) = \frac{b}{20}\\) can be used to determine the length of line segment ac.
what is the length of \\(\overline{ac}\\)? round to the nearest tenth.
15.3 cm
11.8 cm
14.9 cm
12.9 cm

Explanation:

Step1: Analyze the right triangle

In right triangle \(ABC\) with \(\angle C = 90^{\circ}\), \(\angle B=40^{\circ}\), and hypotenuse \(AB = 20\space\text{cm}\). We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle B\), the opposite side to \(\angle B\) is \(AC\) and hypotenuse is \(AB\). So the given equation is \(\sin(40^{\circ})=\frac{AC}{20}\).

Step2: Solve for \(AC\)

To find \(AC\), we can rearrange the equation: \(AC = 20\times\sin(40^{\circ})\). We know that \(\sin(40^{\circ})\approx0.6428\). Then \(AC=20\times0.6428 = 12.856\space\text{cm}\approx12.9\space\text{cm}\) (rounded to the nearest tenth).

Answer:

\(12.9\space\text{cm}\)