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Fixed point: sin x + 3x - 2

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.

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

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)

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

nx(n)x(n+1) = (2 − sin x(n))/3abs diff |x(n+1) − x(n)|
00.5000000.5068580.006858
10.5068580.5048560.002002
20.5048560.5054400.000584
30.5054400.5052690.000170
40.5052690.5053190.000050
50.5053190.5053040.000014
60.5053040.5053090.000004
70.5053090.5053070.000001
80.5053070.5053080.000000

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).

✓ Answer checked by Claude Opus 5.5 (high effort)

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