Question

determine the remaining sides and angles of the triangle abc.
what is the measure of angle b?
48.2°
(simplify your answer. type an integer or a decimal.)
what is the length of side a?
39.7 ft
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

Explanation:

Step1: Find angle \( B \)

In a triangle, the sum of angles is \( 180^{\circ} \). Given \( \angle A=25.9^{\circ} \) and \( \angle C = 90^{\circ} \).

$$ \angle B=180^{\circ}-\angle A - \angle C $$

$$ \angle B=180^{\circ}-25.9^{\circ}-90^{\circ}=64.1^{\circ} $$

Step2: Find side \( a \)

We know that \( \sin A=\frac{\text{opposite}}{\text{hypotenuse}} \). Here, \( \angle A = 25.9^{\circ} \), hypotenuse \( c = 31.1\) ft and opposite side to \( \angle A\) is \( a \).

$$ \sin(25.9^{\circ})=\frac{a}{31.1} $$

$$ a = 31.1\times\sin(25.9^{\circ}) $$

Using a calculator, \( \sin(25.9^{\circ})\approx0.437 \)

$$ a=31.1\times0.437\approx13.6 $$

Answer:

Angle \( B = 64.1^{\circ} \), side \( a\approx13.6\) ft