Solve \( 13 \cos \theta-8=0 \) for \( 0^{\circ} \leq \theta<180^{\circ} \). Give your answer correct to one decimal place.

To solve the equation \( 13 \cos \theta - 8 = 0 \), we start by isolating \( \cos \theta \): \[ 13 \cos \theta = 8 \] \[ \cos \theta = \frac{8}{13} \] Now, we need to find \( \theta \) such that \( 0^{\circ} \leq \theta < 180^{\circ} \). Since \( \cos \theta \) is positive, we will have two possible angles in this range: one in the first quadrant and one in the second quadrant. 1. For the first quadrant: \[ \theta_1 = \cos^{-1}\left(\frac{8}{13}\right) \] Using a calculator: \[ \theta_1 \approx 43.6^{\circ} \] 2. For the second quadrant: \[ \theta_2 = 180^{\circ} - \theta_1 \] Calculating this gives: \[ \theta_2 \approx 180^{\circ} - 43.6^{\circ} \approx 136.4^{\circ} \] Therefore, the two solutions to \( 13 \cos \theta - 8 = 0 \) in the range \( 0^{\circ} \leq \theta < 180^{\circ} \) are approximately: \[ \theta \approx 43.6^{\circ}, \quad 136.4^{\circ} \] So, the final answer, rounded to one decimal place, is: \[ \theta \approx 43.6^{\circ}, 136.4^{\circ} \]