---
title: "Half range sine series: π − x"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:half-range-sine-series-x"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-x
site: BCT Engineering
---

# Half range sine series: π − x

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Kartik (new course), 2 marks; 2082 Shrawan, 5 marks; 2075 Asoj, 5 marks; 2071 Shrawan, 5 marks (asked 4 times)
- Video: [Half Range Fourier Sine and Cosine Series Example | Lecture II by GP Sir](https://www.youtube.com/watch?v=Sq2FhCxcyI8) — Dr.Gajendra Purohit
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-x
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Obtain the half-range Fourier sine series for f(x) = π − x in the range 0 < x < π.

## Short answer

For the half-range sine series on (0,π), bn = (2/π)∫_0^π (π-x) sin nx dx = 2/n. Thus π-x = 2 Σ_{n=1}^∞ (sin nx)/n, 0<x<π.

## Answer

For a half-range sine series on (0, π):

```text
f(x) = Σ bₙ sin nx,    bₙ = (2/π) ∫₀^π f(x) sin nx dx
```

For f(x) = π − x:

```text
bₙ = (2/π) ∫₀^π (π − x) sin nx dx
```

Integrate by parts: ∫₀^π (π − x) sin nx dx = [−(π − x) cos nx / n]₀^π − ∫₀^π cos nx/n dx = (π/n) − 0 = π/n (the last integral is [sin nx/n²]₀^π = 0).

```text
bₙ = (2/π)(π/n) = 2/n
```

**Answer:**

```text
π − x = 2 Σ_{n=1}^∞ (sin nx)/n = 2 [ sin x + (sin 2x)/2 + (sin 3x)/3 + ... ],  0 < x < π
```

(Check: at x = π/2, π/2 = 2[1 − 1/3 + 1/5 − ...], i.e. π/4 = 1 − 1/3 + 1/5 − ... ✓.)

## Check yourself

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

1. The half-range sine coefficient on (0, pi) is
   A. bn = (1/pi) integral of f(x) sin nx from 0 to pi
   B. bn = (2/pi) integral of f(x) cos nx
   C. bn = (2/pi) integral of f(x) sin nx
   D. bn = (1/2pi) integral
   Answer: C. Factor 2/pi for half-range. (Key point: Formula)

2. integral from 0 to pi of (pi - x) sin nx dx =
   A. 2pi/n
   B. 0
   C. -pi/n
   D. pi/n
   Answer: D. By parts: boundary term gives pi/n, cos integral 0. (Key point: Integral)

3. bn =
   A. 2/n
   B. 2/(pi n)
   C. 1/n
   D. 4/n
   Answer: A. (2/pi)(pi/n). (Key point: bn)

4. Putting x = pi/2 gives
   A. pi/2 = 1 + 1/3 + 1/5
   B. pi/4 = 1 - 1/3 + 1/5 - ...
   C. pi = 1 - 1/2 + 1/3
   D. pi/4 = 1 + 1/3 + 1/5
   Answer: B. sin(n pi/2) is 0 for even n, alternating +-1 for odd n. (Key point: Check)

## Close variants set in other exams

- [Half range sine series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e) — Mathematics III, chapter 4, asked 12×
- [Half range sine series: πx − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2-four) — Mathematics III, chapter 4, asked 12×
- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2-two) — Mathematics III, chapter 4, asked 9×
- [Half range sine series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2) — Mathematics III, chapter 4, asked 14×
- [Half range sine series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2-three) — Mathematics III, chapter 4, asked 11×

## Related in Mathematics III

- [Half range sine series: x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-three) — Mathematics III, chapter 4, asked 6×
- [Half range sine series: e^(ax)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-ax) — Mathematics III, chapter 4, asked 5×
- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2-three) — Mathematics III, chapter 4, asked 9×
- [Half range sine series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-two) — Mathematics III, chapter 4, asked 6×
- [Half range sine series: e^(2x)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-2x) — Mathematics III, chapter 4, asked 6×
- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2) — Mathematics III, chapter 4, asked 3×

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