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Trigonometry delves into the relationships between triangle angles and sides, focusing on sine, cosine, and tangent functions. These functions exhibit periodic behavior and are graphically represented to aid in solving equations. The sine and cosine functions oscillate between -1 and 1, while the tangent function's range is all real numbers. Understanding their graphical properties is crucial for finding solutions within specific intervals.

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## Trigonometric Functions

### Sine Function

The sine function relates the size of an angle to the ratios of a triangle's sides and is characterized by its periodic, sinusoidal graph

### Cosine Function

The cosine function is similar to the sine function but is phase-shifted and exhibits symmetry properties, making it useful for solving equations

### Tangent Function

The tangent function has a unique graph with a period of π radians and an unbounded range, making it useful for solving equations with multiple solutions

## Graphical Properties

### Symmetry

The sine and cosine functions exhibit symmetry properties, allowing for the identification of additional solutions to equations

### Periodicity

Trigonometric functions are inherently periodic, meaning their values repeat in regular intervals, which is crucial for solving equations

### Vertical Asymptotes

The tangent function has vertical asymptotes at odd multiples of π/2 radians, where it approaches infinity, making it useful for solving equations with unbounded solutions

## Solving Trigonometric Equations

### Graphical Methods

Graphical methods involve plotting the trigonometric function and the equation's constant value on the same coordinate plane to identify solutions

### Symmetry Properties

Symmetry properties of trigonometric functions can be leveraged to identify additional solutions to equations

### Periodic Behavior

Understanding the periodic nature of trigonometric functions is essential for accurately solving equations within specific ranges

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