---
title: "Maclaurin expansion of log(1+sin x)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:maclaurin-log-1-plus-sin-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/maclaurin-log-1-plus-sin-x
site: BCT Engineering
---

# Maclaurin expansion of log(1+sin x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2072 Chaitra
- Video: [#05 || Expand log(1+sinx) in power of x by Maclaurin's Series⁉️ || 📄PYQ...||MACLAURIN'S SERIES || ](https://www.youtube.com/watch?v=H9y1PHa0TL0) — Rajan Mathematics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/maclaurin-log-1-plus-sin-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Assuming the validity of expansion, expand log(1+sin x) by Maclaurin's theorem.

## Short answer

Using Maclaurin's theorem: f(0)=0, f'(0)=1, f''(0)=-1, f'''(0)=1, f''''(0)=-2. Thus log(1+sin x) = x - x^2/2 + x^3/6 - x^4/12 + ...

## Answer

Let f(x) = log(1 + sin x). Use Maclaurin's series f(x) = f(0) + x f′(0) + (x²/2!)f″(0) + ...

```text
f′  = cos x/(1 + sin x)
f″  = [−sin x(1+sin x) − cos²x]/(1+sin x)² = −1/(1 + sin x)
f‴  = cos x/(1 + sin x)²
f⁗  = [−sin x(1+sin x) − 2cos²x]/(1+sin x)³
```
At x = 0:

```text
f(0) = 0,  f′(0) = 1,  f″(0) = −1,  f‴(0) = 1,  f⁗(0) = −2
```
Therefore

f(x) = 0 + x − x²/2! + x³/3! − 2x⁴/4! + ...

**Answer: log(1 + sin x) = x − x²/2 + x³/6 − x⁴/12 + ...**

(Check: substituting u = sin x = x − x³/6 in log(1+u) = u − u²/2 + u³/3 − u⁴/4 gives the same series.)

## Check yourself

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

1. f' for f = log(1 + sin x) is:
   A. cos x/(1 + sin x)
   B. 1/(1+sin x)
   C. -cos x/(1+sin x)
   D. sin x/(1+sin x)
   Answer: A. Chain rule. (Key point: Derivative)

2. f'' simplifies to:
   A. -cos x
   B. 1/(1+sin x)
   C. -1/(1 + sin x)
   D. -1/(1 - sin x)
   Answer: C. Numerator -sin x - sin^2 x - cos^2 x = -(1 + sin x). (Key point: Second derivative)

3. f(0), f'(0), f''(0), f'''(0) are:
   A. 0, 1, 1, 1
   B. 0, 0, -1, 1
   C. 0, 1, -1, 1
   D. 1, 0, -1, 1
   Answer: C. Evaluate at 0. (Key point: Values)

4. The series is:
   A. x + x^2/2 + x^3/6
   B. x - x^2 + x^3
   C. x - x^2/2 + x^3/6 - x^4/12
   D. x - x^2/2 + x^3/3
   Answer: C. -2/4! = -1/12. (Key point: Series)

## Related in Mathematics I

- [Maclaurin expansion of e^x log(1+x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Amaclaurin-ex-log-1-plus-x) — Mathematics I, chapter 1
- [Maclaurin expansion of e^(sin x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Amaclaurin-e-sin-x) — Mathematics I, chapter 1
- [Maclaurin expansion of eˣ/(1+eˣ)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Amaclaurin-ex-over-1-plus-ex) — Mathematics I, chapter 1, asked 2×
- [DUIS: ∫ log((a+b sin x)/(a−b sin x)) dx/sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-log-a-plus-b-sinx) — Mathematics I, chapter 2, asked 9×
- [Show ∫ x log(sin x) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aintegral-x-log-sin-x) — Mathematics I, chapter 2
- [Solve (x²D²+xD+1)y=sin(log x²)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-cauchy-x2d2-xd-1-sin-log-x2) — Mathematics I, chapter 4

## Related in other subjects

- [Gauss 2-point and 3-point: log(x+1)+sin 2x](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Agauss-2pt-3pt-log-sin) — Numerical Method, chapter 5
- [Sampling theorem; Nyquist of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 5×
- [Residue theorem: ∮sin z / z⁶ on |z|=1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-sinz-over-z6) — Applied Mathematics, chapter 1, asked 3×
- [Euler's theorem statement with sin u example](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aeuler-s-theorem-statement-sin) — Mathematics, chapter 1, asked 3×