Calculus II · Unit 4B · lesson
Small-Angle Approximations in Physics
Derive small-angle approximations from Taylor series, quantify error, and apply them carefully to elementary physics models.
Section overview
Taylor bounds and approximationWhat this section is building
Derive small-angle approximations from Taylor series, quantify error, and apply them carefully to elementary physics models.
The polynomial supplies the estimate; the remainder theorem supplies the trust boundary.
Choose a tractable center and degree, bound the needed derivative on the whole interval, then compare the bound with the required tolerance.
Evaluating the next term without checking that the theorem's hypotheses make it a valid error bound.
Learning objectives
use Taylor polynomials to justify common small-angle approximations and estimate their errors.
Small-Angle Approximations in Physics
Physical models often simplify because a variable is small
Near zero, and 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 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.
A local approximation can simplify a physical law
For angles near zero measured in radians,
so . 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.
Read this graph as text
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 the line y=theta near zero, with their separation enlarged. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in sine and angle agree locally but separate farther away; color is never the only cue.
Why it matters: Sine and the line y=theta near zero, with their separation enlarged.
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.
Two standard approximations
For small ,
Quantify the sine approximation
At ,
Thus the linear approximation is accurate to better than two ten-thousandths.
Estimate the pendulum-model error at ten degrees
Ten degrees is radians. The leading difference between and is approximately
Relative to , this is about . Thus the small-angle replacement is quite accurate at ten degrees, while the calculation also quantifies what “small” means.
Small-angle formulas require radians
The approximation compares dimensionless radian measure with sine. Substituting degrees directly can produce nonsense.
u4b-small_angle_physics-01Bound the error in using the first omitted term.
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The first omitted sine term is .
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Find an error bound for .
Determine the largest angle making the sine linearization error below .
Explain why radians are required.
Describe how the approximation changes the pendulum equation.
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