An approximate value of `cos 40^(@)` is

Options

A. 0.7688
B. 0.7071
C. 0.7117
D. 0.7

Answer

To find an approximate value of \( \cos 40^\circ \) using the method of derivatives, we can follow these steps:

### Step 1: Choose a point close to 40 degrees
We know that \( \cos 30^\circ \) is a value we can use because it is close to 40 degrees. Let \( x = 30^\circ \).

### Step 2: Calculate \( \cos 30^\circ \)
The value of \( \cos 30^\circ \) is:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.8660 \]

### Step 3: Determine \( dx \)
Since we are moving from 30 degrees to 40 degrees, we have:
\[ dx = 40^\circ - 30^\circ = 10^\circ \]

### Step 4: Convert \( dx \) to radians
To use the derivative, we need to convert degrees to radians. The conversion is done using:
\[ dx = 10^\circ \times \frac{\pi}{180} = \frac{\pi}{18} \text{ radians} \]

### Step 5: Find the derivative of \( \cos x \)
The derivative of \( \cos x \) is:
\[ \frac{dy}{dx} = -\sin x \]
At \( x = 30^\circ \):
\[ \sin 30^\circ = \frac{1}{2} \]
So,
\[ \frac{dy}{dx} \bigg|_{x=30^\circ} = -\sin 30^\circ = -\frac{1}{2} \]

### Step 6: Calculate \( dy \)
Using the derivative, we can find \( dy \):
\[ dy = \frac{dy}{dx} \cdot dx = -\frac{1}{2} \cdot \frac{\pi}{18} \]
Calculating this gives:
\[ dy = -\frac{\pi}{36} \approx -0.0873 \]

### Step 7: Find \( \cos 40^\circ \)
Now we can approximate \( \cos 40^\circ \):
\[ \cos 40^\circ \approx \cos 30^\circ + dy \]
Substituting the values:
\[ \cos 40^\circ \approx 0.8660 - 0.0873 \approx 0.7787 \]

### Final Result
Thus, the approximate value of \( \cos 40^\circ \) is:
\[ \cos 40^\circ \approx 0.7787 \]