Find the acute angle A . \( \tan \mathrm{A}=2.893 \) \( \mathrm{~A} \approx \square^{\circ} \) (Round to the nearest tenth as needed.)

To find the acute angle \( A \) such that \( \tan A = 2.893 \), we'll use the inverse tangent function (\( \arctan \)). 1. **Understanding the Problem:** - We need to find \( A \) in degrees where \( 0^\circ < A < 90^\circ \). - Given \( \tan A = 2.893 \). 2. **Approach:** - Use the \( \arctan \) function to determine the angle. - Since exact calculation without a calculator is challenging, we'll approximate based on known values. 3. **Approximation Steps:** - **Known Values:** - \( \tan(70^\circ) \approx 2.747 \) - \( \tan(71^\circ) \approx 2.904 \) - \( \tan(72^\circ) \approx 3.077 \) - **Interpolation:** - \( \tan A = 2.893 \) lies between \( \tan(70^\circ) \) and \( \tan(71^\circ) \). - The difference between \( \tan(70^\circ) \) and \( \tan(71^\circ) \) is \( 2.904 - 2.747 = 0.157 \) per degree. - To reach \( \tan A = 2.893 \), the difference needed from \( \tan(70^\circ) \) is \( 2.893 - 2.747 = 0.146 \). - The fractional degree needed is approximately \( 0.146 / 0.157 \approx 0.930 \) degrees. - **Calculation:** - \( A \approx 70^\circ + 0.930^\circ = 70.930^\circ \) - Rounding to the nearest tenth, \( A \approx 70.9^\circ \). 4. **Final Answer:** Rounding to the nearest tenth of a degree, the acute angle \( A \) is approximately: \[ \boxed{70.9^\circ} \]