Question

question
solve for x. round to the nearest tenth, if necessary.
answer attempt 2 out of 2
x =
submit answer

Explanation:

Step1: Identify trigonometric ratio

In right triangle \( \triangle IJK \), \( \angle J = 90^\circ \), \( \angle K = 42^\circ \), and \( IJ = 4.6 \). We need to find \( x = JK \)? Wait, no, \( x \) is \( J K \)? Wait, \( IJ \) is opposite to \( \angle K \), and \( x \) is adjacent? Wait, no, \( IJ = 4.6 \) is opposite to \( \angle K \), and \( x \) is adjacent? Wait, no, \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). Wait, \( \angle K = 42^\circ \), \( IJ \) is opposite to \( \angle K \) (length 4.6), and \( x \) is adjacent to \( \angle K \)? Wait, no, \( J \) is right angle, so \( IJ \) and \( JK \)? Wait, \( IJ = 4.6 \), \( \angle K = 42^\circ \), so \( \tan(42^\circ)=\frac{IJ}{x} \)? Wait, no, \( \tan(\angle K)=\frac{IJ}{x} \), so \( \tan(42^\circ)=\frac{4.6}{x} \)? Wait, no, \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so opposite to \( \angle K \) is \( IJ = 4.6 \), adjacent is \( x \) (since \( J \) is right angle, \( JK \) is adjacent, \( IJ \) is opposite). So \( \tan(42^\circ)=\frac{4.6}{x} \), so \( x=\frac{4.6}{\tan(42^\circ)} \). Wait, or is \( x \) the opposite? Wait, maybe I mixed up. Wait, \( \angle J = 90^\circ \), so sides: \( IJ \) is one leg, \( JK \) is another leg, \( IK \) is hypotenuse. \( \angle K = 42^\circ \), so \( \tan(42^\circ)=\frac{IJ}{JK} \), so \( \tan(42^\circ)=\frac{4.6}{x} \), so \( x = \frac{4.6}{\tan(42^\circ)} \).

Step2: Calculate \( \tan(42^\circ) \)

\( \tan(42^\circ) \approx 0.9004 \)

Step3: Solve for \( x \)

\( x=\frac{4.6}{0.9004} \approx 5.1 \) (rounded to nearest tenth)

Wait, wait, maybe I had it reversed. Wait, if \( \angle K = 42^\circ \), then \( \tan(\angle K) = \frac{\text{opposite}}{\text{adjacent}} = \frac{IJ}{JK} \), so \( JK = \frac{IJ}{\tan(\angle K)} \). So \( IJ = 4.6 \), \( \tan(42^\circ) \approx 0.9004 \), so \( x = \frac{4.6}{0.9004} \approx 5.1 \).

Answer:

\( 5.1 \)