---
title: "False position: e^cos x - sin x - 1"
subject: "Numerical Method"
chapter: 2
key: "semester_4/numerical_method:false-position-ecosx"
source: https://www.bctengineer.com/q/semester_4/numerical_method/false-position-ecosx
site: BCT Engineering
---

# False position: e^cos x - sin x - 1

Numerical Method · Chapter 2: Solutions of Nonlinear Equations

- Asked in: 2070 Bhadra, 6 marks
- Video: [Regula falsi method in 5 minutes](https://www.youtube.com/watch?v=hqlQeWRPyvI) — Smart Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/false-position-ecosx
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-2

## Question

Find a real root of the following equation correct to four decimals using False Position method: e^(cos x) - sin x - 1 = 0

## Short answer

Apply regula falsi to f(x)=e^(cos x)-sin x-1 with a=0, b=1. Iterations converge to c=0.9381, so the real root is approximately 0.9381.

## Answer

**False Position (Regula Falsi) formula:** for f(a)·f(b) < 0,
c = (a·f(b) − b·f(a)) / (f(b) − f(a)). If f(a)·f(c) < 0 set b = c, otherwise set a = c.

#### Step 1: Bracket the root
f(x) = e^(cos x) − sin x − 1  (radians)
f(0) = e − 1 = 1.7183 > 0 and f(1) = 1.7166 − 0.8415 − 1 = −0.1249 < 0, so a root lies in (0, 1).
Take a = 0, b = 1.

#### Step 2: Iterations
| n | a | b | f(a) | f(b) | c | f(c) |
|---|---|---|---|---|---|---|
| 1 | 0.0000 | 1.0000 | 1.7183 | -0.1249 | 0.9322 | 0.0120 |
| 2 | 0.9322 | 1.0000 | 0.0120 | -0.1249 | 0.9382 | -0.0002 |
| 3 | 0.9322 | 0.9382 | 0.0120 | -0.0002 | 0.9381 | -0.0000 |
| 4 | 0.9322 | 0.9381 | 0.0120 | -0.0000 | 0.9381 | -0.0000 |

Successive values of c agree to four decimals (|c(n) − c(n−1)| < 0.00005).

**Answer: root ≈ 0.9381**

## Check yourself

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

1. Root in (0, 1) because:
   A. f(0) = 1.7183 > 0 and f(1) = -0.1249 < 0
   B. f(0) < 0
   C. f(1) > 0
   D. f(0.5) = 0
   Answer: A. Sign change. (Key point: Bracket)

2. First c is computed by:
   A. (a+b)/2
   B. c = (a f(b) - b f(a))/(f(b) - f(a))
   C. a - f(a)/f'(a)
   D. a*b
   Answer: B. Chord intersection. (Key point: Formula)

3. After c = 0.9322 with f(c) > 0 (f(a) > 0), the algorithm sets:
   A. b = c
   B. Stops
   C. a = c
   D. a = 0
   Answer: C. f(a)*f(c) > 0, so the root is in [c, b]. (Key point: Update)

4. The root is:
   A. 0.9322
   B. 1.0000
   C. 0.0120
   D. 0.9381
   Answer: D. Converges to 0.9381. (Key point: Result)

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