Question

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

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \( \triangle IJK\) with \( \angle I = 90^{\circ}\), we know the side adjacent to \( \angle J\) (\(IJ\) is not needed here as we use \(IK = 2\) (opposite side to \( \angle J\)) and \(JK=3\) (hypotenuse). We use the sine ratio \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \( \sin J=\frac{IK}{JK}\).

Step2: Substitute the values

Substitute \(IK = 2\) and \(JK = 3\) into the sine formula. We get \( \sin J=\frac{2}{3}\).

Step3: Find the angle

To find \(m\angle J\), we use the inverse - sine function. \(J=\sin^{- 1}(\frac{2}{3})\). Using a calculator, \( \sin^{-1}(\frac{2}{3})\approx41.8^{\circ}\) (since \( \sin^{-1}(x)\) gives the angle whose sine is \(x\) and \( \sin^{-1}(\frac{2}{3})\approx41.8103^{\circ}\), rounded to the nearest tenth).

Answer:

\(41.8\)