Estimate cos(1.74)

The table gives sin x, and d/dx (sin x) = cos x. So cos(1.74) is the first derivative of the tabulated function at x = 1.74, found by numerical differentiation (h = 0.04).

Central difference formula

x = 1.74 has a point on each side (1.70 and 1.78), so the central difference is best:

f′(x₀) ≈ [f(x₀ + h) − f(x₀ − h)] / (2h)

cos(1.74) ≈ [sin(1.78) − sin(1.70)] / (2 × 0.04)
          = (0.9781 − 0.9916) / 0.08
          = −0.0135 / 0.08
          = −0.1688

Check with Newton's forward formula (x₀ = 1.70, p = 1)

Forward differences: Δy₀ = −0.0059, Δ²y₀ = −0.0017, Δ³y₀ = 0.0003, Δ⁴y₀ = −0.0006.

f′ = (1/h)[Δy₀ + (2p−1)/2·Δ²y₀ + (3p²−6p+2)/6·Δ³y₀ + (4p³−18p²+22p−6)/24·Δ⁴y₀]
   = 25[−0.0059 + 0.5(−0.0017) + (−1/6)(0.0003) + (2/24)(−0.0006)]
   = 25 × (−0.00685)  = −0.1713

The central value is more reliable, because the truncation error of the one-sided formula is larger and the tabulated sines carry rounding error in the fourth place.

Answer: cos(1.74) ≈ −0.1688 (exact value −0.1684; the negative sign is correct because 1.74 rad > π/2).