Represent smooth functions as polynomials; the basis of small-angle approximations.
Taylor series f(x)=Σ f⁽ⁿ⁾(a)(x-a)ⁿ/n! approximate smooth functions by polynomials; Maclaurin is at a=0. Examples: sin x≈x-x³/6, eˣ≈1+x+x²/2. Truncation error is bounded by the next term.
For a pendulum with small θ, sin θ≈θ gives simple harmonic motion.
A Taylor series turns a complicated function into a polynomial, which is why engineers use small-angle and linear approximations to simplify analysis.
Represent smooth functions as polynomials; the basis of small-angle approximations.
A Taylor series turns a complicated function into a polynomial, which is why engineers use small-angle and linear approximations to simplify analysis.
When the angle is in radians and small, typically below about 0.2 rad or 10 degrees, where the error of sin theta versus theta is under one percent.
Keep enough terms that the next term is smaller than your required accuracy. Check the error by comparing with the exact value at a test point.
Typical applications include replacing sin theta by theta for a small pendulum swing, approximating the exponential term in a thermal model and linearising a sensor nonlinearity near a calibration point.
Study "Sequences convergence tests" first (topic 39 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Small-angle: sin θ≈θ. Radius of convergence matters. Used to linearise and derive numerical methods.
For a pendulum with small θ, sin θ≈θ gives simple harmonic motion.
The next topic is "Partial derivatives", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 40 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
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