Question

solve for ( x ). round to the nearest tenth, if necessary.

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( KLM \) (right-angled at \( L \)), we know the length of the side opposite the \( 25^\circ \) angle (\( KL = 3.1 \)) and we need to find the hypotenuse \( x \) ( \( KM \) ). The sine function relates the opposite side and the hypotenuse in a right triangle, where \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). Here, \( \theta = 25^\circ \), opposite side \( = 3.1 \), and hypotenuse \( = x \). So, \( \sin(25^\circ)=\frac{3.1}{x} \).

Step2: Solve for \( x \)

Rearranging the formula for \( x \), we get \( x = \frac{3.1}{\sin(25^\circ)} \). Calculate \( \sin(25^\circ) \approx 0.4226 \). Then, \( x \approx \frac{3.1}{0.4226} \approx 7.3 \).

Answer:

\( x \approx 7.3 \)