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The small angle approximation is a mathematical method used to simplify trigonometric calculations for angles close to zero. It states that sine, cosine, and tangent of small angles can be approximated by simple algebraic expressions, making it easier to solve problems in physics and engineering. This technique is based on the premise that for small angles measured in radians, these trigonometric functions can be replaced with linear or near-linear terms, thus facilitating quicker and more efficient computations in various scientific and technical fields.

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## Definition and Purpose

### Mathematical technique

The small angle approximation is a mathematical technique used to simplify the calculation of trigonometric functions for small angles measured in radians

### Purpose

The small angle approximation is particularly useful in physics and engineering where small angles frequently occur, and exact trigonometric values are less critical

### Fundamental equations

The small angle approximation is encapsulated by three fundamental equations for the sine, cosine, and tangent functions, which can be approximated by algebraic expressions for small angles

## Derivation and Validity

### Derivation of the small angle approximation for cosine

The small angle approximation for cosine is derived from the Pythagorean identity and the small angle approximation for sine

### Validity of the approximations

The small angle approximations are valid for angles where the angle is sufficiently small, typically less than about 0.1 radians

### Graphical representation

The small angle approximations can be visualized graphically by plotting the functions and observing their close convergence near the origin

## Applications

### Problem-solving

The small angle approximation is invaluable in solving problems involving small angles, as it simplifies complex calculations in fields such as physics and engineering

### Conversion to radians

It is crucial to use radians when applying the small angle approximation, as angles given in degrees must be converted to radians for accurate results

### Examples

Examples of using the small angle approximation include simplifying expressions such as \(\frac{\cos \theta}{\sin \theta}\) and \(\tan (3x) \cos (2x)\) for small angles

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