---
title: "Estimate cos(1.74)"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:interpolate-cos-1-74"
source: https://www.bctengineer.com/q/semester_4/numerical_method/interpolate-cos-1-74
site: BCT Engineering
---

# Estimate cos(1.74)

Numerical Method · Chapter 4: Interpolation

- Asked in: 2073 Bhadra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/interpolate-cos-1-74
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

Estimate the value of cos(1.74) from the following data:
| x | 1.7 | 1.74 | 1.78 | 1.82 | 1.86 |
|---|---|---|---|---|---|
| Sin(x) | 0.9916 | 0.9857 | 0.9781 | 0.9691 | 0.9584 |

## Short answer

Since cos x is the derivative of sin x, use the central difference: cos(1.74) approx [sin(1.78) - sin(1.70)]/(2*0.04) = -0.0135/0.08 = -0.1688. This central estimate is more reliable; the exact value is -0.1684.

## Answer

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)

```text
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.

```text
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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How can cos(1.74) be estimated from a table of sin x?
   A. Read the sin value at 1.74 directly
   B. Use sin^2 + cos^2 = 1 with the table value only
   C. Treat it as the derivative of sin x at 1.74 using numerical differentiation
   D. Integrate the table
   Answer: C. d/dx sin x = cos x, so cos(1.74) = f'(1.74). (Key point: Link to derivative)

2. Which formula gives f'(x0) at 1.74 with h = 0.04?
   A. [f(1.78) + f(1.70)]/(2h)
   B. f(1.74)/h
   C. [f(1.78) - f(1.74)]/h^2
   D. [f(1.78) - f(1.70)]/(2h)
   Answer: D. The central difference uses the points on both sides. (Key point: Central difference)

3. Why is the central value preferred over the forward-formula value?
   A. It always matches the exact answer
   B. Its truncation error is smaller than the one-sided formula
   C. It uses more terms
   D. It uses cos values
   Answer: B. The one-sided formula has larger truncation error; central gives -0.1688. (Key point: Reliability)

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