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SMALL ANGLE APPROXIMATIONS

In developing linear theories, the small angle assumptions can be a powerful assumption for the engineer. Using small angle assumption one can approximate a trigonometric function with an algebraic function. In many cases this will simplify the problem and permit it to be solved in a closed form or rapid numerical approach. The Taylor series is the key to the small angle assumption.
TAYLOR SERIES
If f(x) is infinitely differentiable in the neighborhood of a point x0, then
(1) f(x0) = ∞⎲⎳n = 0an(x0 − b)n
where the function is defined at a point b, and the coefficients, an are determined by
(2) an = (f(n)(b))/(n!)
The denominator includes n!, which is a mathematical operation known as a factorial. The factorial is computed mathematically using the equation n! = ∏ni = 1i, which is simply a more elegant way to write n(n − 1)(n − 2)...(2)(1).
One can truncate the series for f(x0) at some n = N where N ≥ 1 to have an approximation of f(x0).
Consider the Taylor series for the trigonometric functions of the sine, cosine, and tangent operations:
sin(x) 
(3) sin(x) = x − (x3)/(3!) + (x5)/(5!) + ...
  • where the first-order approximation for sin(x) is x and the third order approximation is x − (x3)/(3!) = x − (x3)/(6).
cos(x) 
(4) cos(x) = 1 − (x2)/(2!) + (x4)/(4!) + ...
  • where the zeroth-order approximation for cos(x) is 1 and the second-order approximation is 1 − (x2)/(2!) = 1 − (x2)/(2)
tan(x) 
(5) tan(x) = x + (x3)/(3!) + (2x5)/(15) + ...
  • where the first-order approximation for tan(x) is x and the third-order approximation is x + (x3)/(3!) = x + (x3)/(6)
The typical linear analysis would thus approximate these trigonometric functions with sin(x) ≈ x, cos(x) ≈ 1, and tan(x) ≈ x. While cosine of a small number is 1, the sine and tangent quantities are not 0. This is an important concept when linearizing or approximating these functions because a zero value can significantly change the result, including indicating a trivial solution rather than a solution of interest.
Some mathematical examples and applications in structures/structural dynamics and aerodynamics are provided in the examples and applications portion of this section.