Question

find ( mangle b ).

write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle b = square^circ )

Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right triangle with right angle at \( C \). We know the opposite side to \( \angle B \) (which is \( AC = 3.2 \)) and the hypotenuse? Wait, no, \( AB \) is the hypotenuse? Wait, no, \( AB = 8.3 \), \( AC = 3.2 \), right angle at \( C \). So for \( \angle B \), the opposite side is \( AC = 3.2 \), and the hypotenuse is \( AB = 8.3 \)? Wait, no, in a right triangle, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). So \( \sin(\angle B)=\frac{AC}{AB}=\frac{3.2}{8.3} \).

Step2: Calculate the angle

First, compute \( \frac{3.2}{8.3}\approx0.3855 \). Then, take the inverse sine (arcsin) of this value: \( \angle B=\arcsin(0.3855) \). Using a calculator, \( \arcsin(0.3855)\approx22.7^\circ \) (rounded to the nearest tenth).

Answer:

\( 22.7 \)