← Taylor Series
θ
Topic 4 of 6

Small-Angle Approximations

The small-angle approximation — replacing sin θ with θ — is arguably the most-used approximation in physics. It converts nonlinear differential equations into solvable linear ones. This topic covers when to use it, what the error is, and the important consequences like the simple pendulum period.

Key Concepts

Small-Angle Approximation
For θ≪1\theta \ll 1 radian: sin⁡θ≈θ\sin\theta \approx \theta, cos⁡θ≈1−θ2/2\cos\theta \approx 1 - \theta^2/2, and tan⁡θ≈θ\tan\theta \approx \theta. These are just the leading terms of the Maclaurin series.
Linearization
Replacing sin⁡θ\sin\theta with θ\theta converts the nonlinear pendulum equation θ¨+(g/L)sin⁡θ=0\ddot{\theta} + (g/L)\sin\theta = 0 into the linear harmonic oscillator θ¨+(g/L)θ=0\ddot{\theta} + (g/L)\theta = 0, which has an exact closed-form solution.
Fractional Error
The fractional error from using sin⁡θ≈θ\sin\theta \approx \theta is ∣sin⁡θ−θ∣/sin⁡θ≈θ2/6|\sin\theta - \theta|/\sin\theta \approx \theta^2/6. At θ=10°≈0.175\theta = 10° \approx 0.175 rad, the error is about 0.5%.
Range of Validity
The approximation is typically considered valid for θ≲0.2\theta \lesssim 0.2 rad (about 11°), where the error in sin⁡θ≈θ\sin\theta \approx \theta is under 1%.

Key Equations

Small-Angle Series
sin⁡θ≈θ−θ36,cos⁡θ≈1−θ22,tan⁡θ≈θ+θ33\sin\theta \approx \theta - \frac{\theta^3}{6}, \quad \cos\theta \approx 1 - \frac{\theta^2}{2}, \quad \tan\theta \approx \theta + \frac{\theta^3}{3}
Leading terms of the Maclaurin series, valid for small θ\theta in radians.
Simple Pendulum Period
T=2πLgT = 2\pi\sqrt{\frac{L}{g}}
Valid only in the small-angle limit. The exact period involves an elliptic integral.
Period Correction
T≈T0 ⁣(1+θ0216)T \approx T_0\!\left(1 + \frac{\theta_0^2}{16}\right)
First correction for amplitude θ0\theta_0. The period grows with amplitude — a fact the small-angle formula hides.
Worked Example

Period of a Simple Pendulum

Problem

A pendulum has length L=1.5L = 1.5 m. Using the small-angle approximation, find the period TT and the fractional error in period caused by using this approximation at amplitude θ0=0.3\theta_0 = 0.3 rad.

Solution

The small-angle period: T0=2πL/g=2π1.5/9.8T_0 = 2\pi\sqrt{L/g} = 2\pi\sqrt{1.5/9.8}.

T0=2π0.1531=2π×0.3913=2.459 sT_0 = 2\pi\sqrt{0.1531} = 2\pi \times 0.3913 = 2.459 \text{ s}

The fractional error in period at amplitude θ0=0.3\theta_0 = 0.3 rad:

ΔTT0≈θ0216=(0.3)216=0.0916=0.005625\frac{\Delta T}{T_0} \approx \frac{\theta_0^2}{16} = \frac{(0.3)^2}{16} = \frac{0.09}{16} = 0.005625

So the true period is about 0.56%0.56\% longer than the small-angle prediction — a small but measurable effect.

Answer T0=2.459T_0 = 2.459 s; fractional period error ≈0.56%\approx 0.56\% at θ0=0.3\theta_0 = 0.3 rad
Practice

Exercises

6 problems
1 of 6

A pendulum of length L=2.0L = 2.0 m. Using the small-angle formula T=2πL/gT = 2\pi\sqrt{L/g} with g=9.8g = 9.8 m/s², what is the period TT in seconds? Round to 3 decimal places.

s
2 of 6

The fractional error in using sin⁡θ≈θ\sin\theta \approx \theta is approximately θ2/6\theta^2/6. For θ=0.15\theta = 0.15 rad, what is this fractional error?

3 of 6

Using cos⁡θ≈1−θ2/2\cos\theta \approx 1 - \theta^2/2, compute the approximation of cos⁡(0.2 rad)\cos(0.2\text{ rad}).

Unlock Exercise 3

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →
4 of 6

The small-angle approximation is valid to within 1% when θ2/6<0.01\theta^2/6 < 0.01. What is the maximum θ\theta (in radians) for 1% accuracy? Round to 3 decimal places.

Unlock Exercise 4

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →
5 of 6

A pendulum of length L=0.5L = 0.5 m. What is its small-angle period T0T_0 in seconds? Use g=9.8g = 9.8 m/s².

Unlock Exercise 5

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →
6 of 6

For amplitude θ0=0.4\theta_0 = 0.4 rad, the fractional increase in pendulum period over the small-angle value is θ02/16\theta_0^2/16. What is this value?

Unlock Exercise 6

Subscribe to PhysWeb Pro to access all exercises and track your progress.

Upgrade to Pro →

Key Takeaways

  • For small θ\theta in radians: sin⁡θ≈θ\sin\theta \approx \theta, cos⁡θ≈1−θ2/2\cos\theta \approx 1 - \theta^2/2. These are the leading terms of the Maclaurin series.
  • The fractional error in sin⁡θ≈θ\sin\theta \approx \theta is ≈θ2/6\approx \theta^2/6. At 10° (0.1750.175 rad) the error is 0.5%0.5\%.
  • The simple pendulum period T=2πL/gT = 2\pi\sqrt{L/g} follows from the small-angle approximation and is exact only in that limit.
  • The true period grows with amplitude: T≈T0(1+θ02/16)T \approx T_0(1 + \theta_0^2/16). The small-angle formula underestimates the period.
  • Linearization is the key technique: replacing sin⁡θ→θ\sin\theta \to \theta converts nonlinear differential equations into tractable linear ones.
← PreviousConvergence & Error Bounds