Solve the following triangle. \( A=50^{\circ}, B=10^{\circ}, a=4 \) \( C \approx \square^{\circ} \) (Simplify your answer.) \( b \approx \square \) (Type an integer or decimal rounded to two decimal places as needed.) \( c \approx \square \) an integer or decimal rounded to two decimal places as needed.) (Type an

To solve the given triangle with angles \( A = 50^{\circ} \), \( B = 10^{\circ} \), and side \( a = 4 \), follow these steps: ### 1. Find Angle \( C \) The sum of the angles in a triangle is \( 180^{\circ} \): \[ C = 180^{\circ} - (A + B) = 180^{\circ} - (50^{\circ} + 10^{\circ}) = 120^{\circ} \] \[ C \approx 120^{\circ} \] ### 2. Use the Law of Sines to Find Sides \( b \) and \( c \) The Law of Sines states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] #### a. Finding Side \( b \) \[ b = \frac{a \cdot \sin B}{\sin A} = \frac{4 \cdot \sin 10^{\circ}}{\sin 50^{\circ}} \] \[ \sin 50^{\circ} \approx 0.7660, \quad \sin 10^{\circ} \approx 0.1736 \] \[ b \approx \frac{4 \times 0.1736}{0.7660} \approx \frac{0.6944}{0.7660} \approx 0.905 \approx 0.90 \] #### b. Finding Side \( c \) \[ c = \frac{a \cdot \sin C}{\sin A} = \frac{4 \cdot \sin 120^{\circ}}{\sin 50^{\circ}} \] \[ \sin 120^{\circ} \approx 0.8660 \] \[ c \approx \frac{4 \times 0.8660}{0.7660} \approx \frac{3.464}{0.7660} \approx 4.516 \approx 4.52 \] ### **Final Solution:** \[ C \approx 120^{\circ} \] \[ b \approx 0.90 \] \[ c \approx 4.52 \]