\( 1 \leftarrow \begin{array}{l}\text { Solve the triangle. } \\ a=9, c=4, B=58^{\circ} \\ b=\square \text { (Round to the nearest tenth as needed.) }\end{array} \)

To solve the triangle with the given conditions \( a = 9 \), \( c = 4 \), and \( B = 58^\circ \), we can use the Law of Sines. The Law of Sines states that: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] ### Step 1: Find angle \( A \) First, we need to find angle \( A \) using the Law of Sines. We can rearrange the formula to find \( A \): \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] However, we need to find angle \( C \) first. We can use the Law of Sines to find \( C \) as follows: \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] Rearranging gives us: \[ \sin C = \frac{c \cdot \sin B}{a} \] Substituting the known values: \[ \sin C = \frac{4 \cdot \sin(58^\circ)}{9} \] ### Step 2: Calculate \( \sin C \) Let's calculate \( \sin(58^\circ) \) and then find \( \sin C \): \[ \sin(58^\circ) \approx 0.8480 \] Now substituting this value: \[ \sin C = \frac{4 \cdot 0.8480}{9} \] Calculating this gives: \[ \sin C \approx \frac{3.392}{9} \approx 0.3769 \] ### Step 3: Find angle \( C \) Now we can find angle \( C \): \[ C = \arcsin(0.3769) \] ### Step 4: Calculate angle \( C \) Let's calculate \( C \): \[ C \approx 22.1^\circ \] ### Step 5: Find angle \( A \) Now we can find angle \( A \) using the fact that the sum of angles in a triangle is \( 180^\circ \): \[ A = 180^\circ - B - C \] Substituting the known values: \[ A = 180^\circ - 58^\circ - 22.1^\circ \] Calculating this gives: \[ A \approx 99.9^\circ \] ### Step 6: Find side \( b \) Now we can find side \( b \) using the Law of Sines: \[ \frac{b}{\sin B} = \frac{a}{\sin A} \] Rearranging gives: \[ b = \frac{a \cdot \sin B}{\sin A} \] Substituting the known values: \[ b = \frac{9 \cdot \sin(58^\circ)}{\sin(99.9^\circ)} \] Since \( \sin(99.9^\circ) \approx \sin(80^\circ) \approx 0.9848 \): Calculating \( b \): \[ b = \frac{9 \cdot 0.8480}{0.9848} \] Calculating this gives: \[ b \approx \frac{7.632}{0.9848} \approx 7.7 \] ### Final Answer Thus, the length of side \( b \) is approximately: \[ \boxed{7.7} \]