`cos 61^(@)`, it being given that `sin 60^(@) = 0.86603 and 1^(@) = 0.01745` radian.

Answer

To find the value of \( \cos 61^\circ \) using the given information, we can follow these steps:

### Step 1: Understand the relationship between degrees and radians
We know that \( 1^\circ \) is approximately equal to \( 0.01745 \) radians. Therefore, \( 1^\circ \) can be converted to radians when needed.

### Step 2: Use the cosine function
We know that:
\[ \cos(61^\circ) = \cos(60^\circ + 1^\circ) \]
We can use the cosine addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
In our case, \( a = 60^\circ \) and \( b = 1^\circ \).

### Step 3: Substitute values
From the problem, we have:
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \sin(60^\circ) = 0.86603 \)
- \( \cos(1^\circ) \) and \( \sin(1^\circ) \) need to be approximated.

Using small angle approximations:
\[ \cos(1^\circ) \approx 1 \quad \text{and} \quad \sin(1^\circ) \approx 0.01745 \]

### Step 4: Apply the cosine addition formula
Now substituting these values into the cosine addition formula:
\[ \cos(61^\circ) = \cos(60^\circ) \cos(1^\circ) - \sin(60^\circ) \sin(1^\circ) \]
\[ \cos(61^\circ) = \left(\frac{1}{2}\right)(1) - (0.86603)(0.01745) \]

### Step 5: Calculate the values
Calculating the second term:
\[ 0.86603 \times 0.01745 \approx 0.0151 \]
Now substituting back:
\[ \cos(61^\circ) = \frac{1}{2} - 0.0151 \]
\[ \cos(61^\circ) = 0.5 - 0.0151 = 0.4849 \]

### Final Result
Thus, the value of \( \cos 61^\circ \) is approximately:
\[ \cos 61^\circ \approx 0.4849 \]

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