- 2070 Bhadra · 6 marks
False position: e^cos x - sin x - 1
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.
Find a real root of the following equation correct to four decimals using False Position method: e^(cos x) - sin x - 1 = 0
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
✓ Answer checked by Claude Opus 5.5 (high effort)
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