Using differential find the approximate value of `tan 46^(@)` , If it is being given that `1^(@) = 0.01745` radian.

Answer

To find the approximate value of \( \tan 46^\circ \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = f(x) = \tan x \).

### Step 2: Identify the point of approximation
We will approximate \( \tan 46^\circ \) around a point where we know the value. We can choose \( x = 45^\circ \) (which is \( \frac{\pi}{4} \) radians) because \( \tan 45^\circ = 1 \).

### Step 3: Calculate \( \Delta x \)
Since we want to find \( \tan 46^\circ \), we have:
\[ \Delta x = 46^\circ - 45^\circ = 1^\circ \]
Now, we need to convert degrees to radians:
\[ \Delta x = 1^\circ = 1 \times \frac{\pi}{180} \approx 0.01745 \text{ radians} \]

### Step 4: Find the derivative
The derivative of \( y = \tan x \) is:
\[ \frac{dy}{dx} = \sec^2 x \]
Now we will evaluate the derivative at \( x = 45^\circ \):
\[ \frac{dy}{dx} \bigg|_{x = 45^\circ} = \sec^2(45^\circ) = 2 \]

### Step 5: Use the differential to approximate \( \Delta y \)
Using the formula for differentials:
\[ dy = \frac{dy}{dx} \cdot dx \]
Substituting the values we have:
\[ dy = 2 \cdot 0.01745 \approx 0.03490 \]

### Step 6: Calculate \( \tan 46^\circ \)
Now we can approximate \( \tan 46^\circ \) using:
\[ \tan 46^\circ \approx \tan 45^\circ + dy \]
Substituting the known values:
\[ \tan 46^\circ \approx 1 + 0.03490 \approx 1.03490 \]

### Final Answer
Thus, the approximate value of \( \tan 46^\circ \) is:
\[ \tan 46^\circ \approx 1.03490 \]