Find the approximate values of :
`tan(44^(@))," given "1^(@)=0.0175^( c )`.

Answer

To find the approximate value of \( \tan(44^\circ) \) given that \( 1^\circ = 0.0175 \) radians, we can use the concept of derivatives and linear approximation.

### Step-by-Step Solution:

1. **Identify the Function and the Point of Approximation**:
We know that \( \tan(x) \) is the function we want to approximate. We will use the point \( x = 45^\circ \) for our approximation since it is close to \( 44^\circ \) and we know \( \tan(45^\circ) = 1 \).

2. **Express \( 44^\circ \) in Terms of \( 45^\circ \)**:
We can express \( 44^\circ \) as:
\[ 44^\circ = 45^\circ - 1^\circ \]
Here, we set \( x = 45^\circ \) and \( \Delta x = -1^\circ \).

3. **Use the Linear Approximation Formula**:
The linear approximation formula is given by:
\[ f(x + \Delta x) \approx f(x) + f'(x) \Delta x \]
In our case, \( f(x) = \tan(x) \), \( x = 45^\circ \), and \( \Delta x = -1^\circ \).

4. **Calculate the Derivative**:
The derivative of \( f(x) = \tan(x) \) is:
\[ f'(x) = \sec^2(x) \]
At \( x = 45^\circ \):
\[ f'(45^\circ) = \sec^2(45^\circ) = 2 \]

5. **Substitute Values into the Approximation Formula**:
Now substituting into the linear approximation:
\[ \tan(44^\circ) \approx \tan(45^\circ) + \tan'(45^\circ)(-1^\circ) \]
\[ \tan(44^\circ) \approx 1 + 2 \times (-1^\circ) \]

6. **Convert Degrees to Radians**:
Since \( 1^\circ = 0.0175 \) radians, we have:
\[ -1^\circ = -0.0175 \text{ radians} \]
Therefore:
\[ \tan(44^\circ) \approx 1 - 2 \times 0.0175 \]

7. **Calculate the Final Value**:
\[ \tan(44^\circ) \approx 1 - 0.035 = 0.965 \]

### Final Answer:
\[ \tan(44^\circ) \approx 0.965 \]