Find the approximate value of $\cos \left({60}^{\circ }. 5^1\right)$

To calculate the approximate value of cos 60° using a differential approach, follow these steps:

1. Define the Function

Let the function be:

f(x) = cos(x)

The derivative of f(x) is:

f'(x) = -sin(x)

2. Use the Approximation Formula

The formula for finding the approximate value is:

f(x + Δx) ≈ f(x) + f'(x) * Δx

Substitute f(x) = cos 60° and Δx = 5' (5 minutes in radians):

cos(60° + Δx) ≈ cos(60°) - sin(60°) * Δx

3. Convert 5 Minutes to Radians

Since 1 degree = π/180 radians, 5 minutes = (5/60) * π/180:

Δx = (5 / 60) * (π / 180) = 0.0001453 radians

4. Substitute Values

We know:

• cos 60° value is 1/2
• sin 60° value is √3/2

Substitute these values:

    cos(60° + 5') ≈ cos 60° - sin 60° * Δx
                  ≈ (1/2) - (√3/2) * 0.0001453
                  ≈ 0.5 - 0.001258
                  ≈ 0.4987
    

5. Final Answer

Thus, the approximate value of cos 60° for 5 minutes is:

cos 60° + 5' ≈ 0.4987