Small angle approximation: Difference between revisions

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The '''Small angle approximation''' is a rule that says that for small angles, the [[trigonometric function]]s sine and tangent are approximately equal to the angle. This approximation is relevant only when angles are measured in [[radians]]. Of course, the equality is not exact; only when the angle is zero are the three truly equal. In symbolic terms:
The '''small angle approximation''' is a rule that says that for small angles, the [[trigonometric function]]s sine and tangent are approximately equal to the angle. The approximation is valid only when angles are measured in [[radian]]s. Of course, the equality is not exact except when the angle is zero. In symbolic terms:


<math> \theta \approx \sin \theta \approx \tan \theta </math>
:<math> \theta \approx \sin \theta \approx \tan \theta \!</math>
 
Using the rule, a physical equation such as the equation for [[diffraction]] minima:
 
:<math> b \sin (\theta ) = m \lambda \!</math>
 
might become
 
:<math> b \theta = m \lambda \!</math>.
 
The equation can then be written as a pure product of quantities with whole-numbered exponents, which is sometimes useful.
 
The rule is very useful to an engineer performing experiments or making approximations. It is probably best to avoid it when accuracy is important, or when larger angles are expected.
 
== Pendulum motion ==
 
The small angle approximation is used to solve for the motion of a pendulum when studying oscillatory motion in introductory physics courses. As long as the pendulum string makes a small angle from the vertical, the approximation is valid and simplifies solving the equation of motion for the pendulum.
 
== Mathematical comment ==
 
Mathematically, the small angle approximation is the first-order [[Taylor series|Maclaurin series]] of the sine function about the value zero. Recall Maclaurin's theorem:
 
Let <math>f: R \to R</math> be a function which is ''n'' times differentiable in some proximity of the point zero. The ''Maclaurin polynomial'' order ''n'' is defined as:
 
:<math>f(0) + x \frac{f'(0)}{1!} + x^2 \frac{f''(0)}{2!} + ... + x^n \frac{f^{(n)}(0)}{n!} + (\textrm{remainder}),</math>
 
where the remainder approaches zero as <math>x \to 0</math> and as <math>n \to \infty</math>. We find the first-order Maclaurin approximation as
 
:<math> \sin (0) + x \cos (0) = x. \!</math>
 
This shows how the small angle approximation is arrived at. The approximation can be used for purely mathematical purposes as well:
 
Say that we want to find the limit
 
:<math>\lim_{x \to 0} \quad \frac{\sin x}{x}.</math>
 
Substituting the approximation for sine, we get
 
:<math>\lim_{x \to 0} \quad \frac{x}{x} = 1.</math>
 
We can use the approximation here because the approximation becomes better and better as ''x'' approaches zero.
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Latest revision as of 20:13, 29 January 2022

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The small angle approximation is a rule that says that for small angles, the trigonometric functions sine and tangent are approximately equal to the angle. The approximation is valid only when angles are measured in radians. Of course, the equality is not exact except when the angle is zero. In symbolic terms:

Using the rule, a physical equation such as the equation for diffraction minima:

might become

.

The equation can then be written as a pure product of quantities with whole-numbered exponents, which is sometimes useful.

The rule is very useful to an engineer performing experiments or making approximations. It is probably best to avoid it when accuracy is important, or when larger angles are expected.

Pendulum motion

The small angle approximation is used to solve for the motion of a pendulum when studying oscillatory motion in introductory physics courses. As long as the pendulum string makes a small angle from the vertical, the approximation is valid and simplifies solving the equation of motion for the pendulum.

Mathematical comment

Mathematically, the small angle approximation is the first-order Maclaurin series of the sine function about the value zero. Recall Maclaurin's theorem:

Let be a function which is n times differentiable in some proximity of the point zero. The Maclaurin polynomial order n is defined as:

where the remainder approaches zero as and as . We find the first-order Maclaurin approximation as

This shows how the small angle approximation is arrived at. The approximation can be used for purely mathematical purposes as well:

Say that we want to find the limit

Substituting the approximation for sine, we get

We can use the approximation here because the approximation becomes better and better as x approaches zero.