---
title: "Fixed point: sin x + 3x - 2"
subject: "Numerical Method"
chapter: 2
key: "semester_4/numerical_method:fixed-point-sinx-3x-2"
source: https://www.bctengineer.com/q/semester_4/numerical_method/fixed-point-sinx-3x-2
site: BCT Engineering
---

# Fixed point: sin x + 3x - 2

Numerical Method · Chapter 2: Solutions of Nonlinear Equations

- Asked in: 2074 Bhadra, 6 marks; 2070 Magh, 6 marks (asked 2 times)
- Video: [SNM| MA3251 | Unit 3 | Fixed point iteration method | Find the real root of the equation cosx=3x–1](https://www.youtube.com/watch?v=hKI78VqWr_g) — Mathematics Kala
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/fixed-point-sinx-3x-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-2

## Question

Find a real root of the equation sin x + 3x - 2 = 0, correct to six decimals, using the Fixed Point iteration method.

## Short answer

Express as x = (2 - sin x)/3, with |g'(x)|≤1/3 ensuring convergence. Starting from x0=0.5, iteration yields root 0.505308 after 8 iterations.

## Answer

**Fixed point method:** write f(x) = 0 as x = g(x) and iterate x(n+1) = g(x(n)). It converges if |g′(x)| < 1 near the root.

#### Step 1: Locate the root
f(x) = sin x + 3x − 2 (x in radians).
f(0) = −2 < 0 and f(1) = 0.8415 + 3 − 2 = 1.8415 > 0, so a root lies in (0, 1). Also f(0.5) = 0.4794 + 1.5 − 2 = −0.0206 < 0, so the root is slightly above 0.5.

#### Step 2: Choose g(x)
```text
3x = 2 − sin x     →     x = g(x) = (2 − sin x) / 3
g'(x) = −cos x / 3,    |g'(x)| ≤ 1/3 < 1  (converges for all x)
```

#### Step 3: Iterate (x0 = 0.5, six decimals)
Formula: x(n+1) = (2 − sin x(n)) / 3

| n | x(n) | x(n+1) = (2 − sin x(n))/3 | abs diff \|x(n+1) − x(n)\| |
|---|---|---|---|
| 0 | 0.500000 | 0.506858 | 0.006858 |
| 1 | 0.506858 | 0.504856 | 0.002002 |
| 2 | 0.504856 | 0.505440 | 0.000584 |
| 3 | 0.505440 | 0.505269 | 0.000170 |
| 4 | 0.505269 | 0.505319 | 0.000050 |
| 5 | 0.505319 | 0.505304 | 0.000014 |
| 6 | 0.505304 | 0.505309 | 0.000004 |
| 7 | 0.505309 | 0.505307 | 0.000001 |
| 8 | 0.505307 | 0.505308 | 0.000000 |

The last two values agree to six decimals (difference < 0.0000005).

**Answer: root ≈ 0.505308** (check: sin x + 3x − 2 ≈ 1.0 × 10⁻⁶ at x = 0.505308).

## Check yourself

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

1. The equation sin x + 3x - 2 = 0 is rewritten as:
   A. x = (2 - sin x)/3
   B. x = 2 - sin x
   C. x = (2 + sin x)/3
   D. x = sin x/3 - 2
   Answer: A. 3x = 2 - sin x. (Key point: Rewrite)

2. Convergence is assured since:
   A. g'(x) = 0
   B. |g'(x)| = |cos x|/3 <= 1/3 < 1
   C. g(x) > 1
   D. |g'(x)| > 1
   Answer: B. The condition |g'| < 1 guarantees convergence. (Key point: Convergence)

3. Root lies in (0,1) because:
   A. f(0) > 0 and f(1) > 0
   B. f(1) = 0
   C. f(0) < 0 and f(1) > 0
   D. g(0) = 0
   Answer: C. There is a sign change. (Key point: Root location)

4. The root to six decimals is:
   A. 0.500000
   B. 0.506858
   C. 1.841471
   D. 0.505308
   Answer: D. Iterates settle at 0.505308. (Key point: Result)

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