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

Question

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

UpStudy AI · Accepted Answer

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:

$$
\boxed{70.9^\circ}
$$
The acute angle $$ A $$ is approximately $$ 70.9^\circ $$.

Find the acute angle A .

(Round to the nearest tenth as needed.)

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Find the acute angle A . (Round to the nearest tenth as needed.)