Calculus II · Unit 4B · lesson

Small-Angle Approximations in Physics

Concept

Learning objectives

use Taylor polynomials to justify common small-angle approximations and estimate their errors.

Small-Angle Approximations in Physics

Explanation

Physical models often simplify because a variable is small

Near zero, $\sin\theta\approx\theta$ and $\cos\theta\approx1-\theta^2/2$ when angles are measured in radians. These are not arbitrary engineering tricks. They are low-degree Maclaurin polynomials whose errors can be bounded.

The approximations linearize otherwise nonlinear models. A pendulum equation involving $\sin\theta$ becomes a linear differential equation when angular displacement is small. Optics, wave motion, and rotational mechanics use similar simplifications. The phrase "small angle" must be tied to a tolerance, not merely intuition.

Bridge

A local approximation can simplify a physical law

For angles near zero measured in radians,

\sin\theta=\theta-\frac{\theta^3}{6}+\cdots,

so $\sin\theta\approx\theta$ . This replacement turns nonlinear equations into linear ones and explains why many small-oscillation systems behave approximately like simple harmonic motion.

The approximation has a domain of usefulness, not universal authority. The cubic correction estimates the error and shows how accuracy deteriorates as the angle grows. Units matter: the derivative-based series is built for radians, so degree input destroys the approximation scale.

**Sine and angle agree locally but separate farther away**
The curves y equals sine theta and y equals theta nearly overlap near zero, while a magnified error panel shows their cubic separation.
Sine and angle agree locally but separate farther away. Sine and the line y=theta near zero, with their separation enlarged.

Concept

Two standard approximations

For small $|\theta|$ ,

\sin\theta=\theta+O(\theta^3),\qquad \cos\theta=1-\frac{\theta^2}{2}+O(\theta^4).

Guided walkthrough

Quantify the sine approximation

At $\theta=0.1$ ,

|\sin\theta-\theta|\le\frac{0.1^3}{3!}\approx1.67\times10^{-4}.

Thus the linear approximation is accurate to better than two ten-thousandths.

Worked example

Estimate the pendulum-model error at ten degrees

Ten degrees is $\theta=\pi/18\approx0.1745$ radians. The leading difference between $\theta$ and $\sin\theta$ is approximately

\frac{\theta^3}{6}\approx\frac{(0.1745)^3}{6}\approx0.000886.

Relative to $\theta$ , this is about $0.51\%$ . Thus the small-angle replacement is quite accurate at ten degrees, while the calculation also quantifies what “small” means.

Common mistake

Small-angle formulas require radians

The approximation $\sin\theta\approx\theta$ compares dimensionless radian measure with sine. Substituting degrees directly can produce nonsense.

Interactive checku4b-small_angle_physics-01

Bound the error in $\sin(0.1)\approx0.1$ using the first omitted term.

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Show hint

The first omitted sine term is $x^3/3!$ .

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Exercise

Find an error bound for $\cos(0.2)\approx1-0.2^2/2$ .

Exercise

Determine the largest angle making the sine linearization error below $10^{-3}$ .

Exercise

Explain why radians are required.

Exercise

Describe how the approximation changes the pendulum equation.

After the explanation

Use the section idea

Reading lens

Pair every Taylor approximation with a degree, center, target input, and certified remainder bound.

Mental model

The polynomial supplies the estimate; the remainder theorem supplies the trust boundary.

Decision

Choose a tractable center and degree, bound the needed derivative on the whole interval, then compare the bound with the required tolerance.

Common trap

Evaluating the next term without checking that the theorem's hypotheses make it a valid error bound.

Check yourself

Can you state exactly why the reported digits are certified?

Source & rights

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