Small angle approximation: Difference between revisions
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imported>Johan Förberg (Added some more reasoning.) |
imported>Johan Förberg (Added thin-negative-spaces to the ends of the formulae; I want that LaTeX rendering!) |
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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 [[radian]]s. 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. This approximation is relevant only when angles are measured in [[radian]]s. Of course, the equality is not exact; only when the angle is zero are the three truly equal. 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: | Using the rule, a physical equation such as the equation for [[diffraction]] minima: | ||
:<math> b \sin (\theta ) = m \lambda </math> | :<math> b \sin (\theta ) = m \lambda \!</math> | ||
might become | might become | ||
:<math> b \theta = m \lambda </math>. | :<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 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. | 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. | ||
Revision as of 04:17, 22 February 2011
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. 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:
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.