Question

the measure of angle a is 13°, and the length of side bc is 4. what are the lengths of the other two sides, rounded to the nearest tenth?
ac =
ab =

Explanation:

Step1: Identify triangle type and trigonometric ratios

This is a right - triangle with right - angle at \(C\), \(\angle A = 13^{\circ}\), and \(BC = 4\). In right - triangle \(ABC\), \(\tan A=\frac{BC}{AC}\) and \(\sin A=\frac{BC}{AB}\).

Step2: Calculate \(AC\)

We know that \(\tan A=\frac{BC}{AC}\), so \(AC = \frac{BC}{\tan A}\). Given \(A = 13^{\circ}\) and \(BC = 4\), \(\tan(13^{\circ})\approx0.2309\). Then \(AC=\frac{4}{\tan(13^{\circ})}=\frac{4}{0.2309}\approx17.3\).

Step3: Calculate \(AB\)

We know that \(\sin A=\frac{BC}{AB}\), so \(AB=\frac{BC}{\sin A}\). \(\sin(13^{\circ})\approx0.2250\). Then \(AB = \frac{4}{\sin(13^{\circ})}=\frac{4}{0.2250}\approx17.8\).

Answer:

\(AC\approx17.3\), \(AB\approx17.8\)