Use differentials, find the approximate value of the following :
`cos""(11pi)/(36)`.

Answer

To find the approximate value of \( \cos\left(\frac{11\pi}{36}\right) \) using differentials, we can follow these steps:

### Step 1: Define the function and the point of approximation
Let \( f(x) = \cos(x) \). We want to approximate \( f\left(\frac{11\pi}{36}\right) \).

### Step 2: Choose a nearby point
We can choose a nearby point where we know the cosine value exactly. A suitable choice is \( x = \frac{\pi}{3} \) because \( \frac{\pi}{3} = \frac{12\pi}{36} \).

### Step 3: Calculate \( \delta x \)
We have:
\[ x + \delta x = \frac{11\pi}{36} \]
Thus,
\[ \delta x = \frac{11\pi}{36} - \frac{\pi}{3} \]
To calculate this, we convert \( \frac{\pi}{3} \) to have a common denominator:
\[ \frac{\pi}{3} = \frac{12\pi}{36} \]
Now substituting:
\[ \delta x = \frac{11\pi}{36} - \frac{12\pi}{36} = -\frac{\pi}{36} \]

### Step 4: Calculate \( \frac{df}{dx} \)
Next, we need the derivative of \( f(x) \):
\[ \frac{df}{dx} = -\sin(x) \]
At \( x = \frac{\pi}{3} \):
\[ \frac{df}{dx} = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]

### Step 5: Calculate \( \delta f \)
Using the formula \( \delta f = \frac{df}{dx} \cdot \delta x \):
\[ \delta f = -\frac{\sqrt{3}}{2} \cdot \left(-\frac{\pi}{36}\right) = \frac{\sqrt{3} \cdot \pi}{72} \]

### Step 6: Calculate \( f\left(\frac{11\pi}{36}\right) \)
Now we can find \( f\left(\frac{11\pi}{36}\right) \):
\[ f\left(\frac{11\pi}{36}\right) \approx f\left(\frac{\pi}{3}\right) + \delta f \]
We know \( f\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ f\left(\frac{11\pi}{36}\right) \approx \frac{1}{2} + \frac{\sqrt{3} \cdot \pi}{72} \]

### Step 7: Numerical approximation
Using \( \pi \approx 3.14 \):
\[ \frac{\sqrt{3}}{2} \approx 0.866 \quad \text{and} \quad \frac{3.14}{72} \approx 0.0436 \]
Thus,
\[ \delta f \approx 0.866 \cdot 0.0436 \approx 0.0378 \]
Finally,
\[ f\left(\frac{11\pi}{36}\right) \approx \frac{1}{2} + 0.0378 \approx 0.5378 \]

### Conclusion
The approximate value of \( \cos\left(\frac{11\pi}{36}\right) \) is approximately \( 0.5378 \).

---