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Understanding the derivatives of basic trigonometric functions is essential in calculus and applied sciences. The sine function's derivative is cosine, while cosine's derivative is negative sine. The tangent function's derivative is secant squared. These derivatives are used to analyze periodic phenomena, such as oscillations in physics, and are proven using limits and differentiation rules like the Chain Rule and the Quotient Rule. Inverse trigonometric functions also have important derivatives for solving calculus problems.

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## Trigonometric Functions

### Sine, Cosine, and Tangent

These functions play a pivotal role in the study of periodic phenomena

### Applications in Physics and Engineering

Simple Pendulums and Electrical Circuits

These functions help model and analyze the behavior of systems ranging from simple pendulums to complex electrical circuits

Velocity and Acceleration of Oscillating Particles

The derivatives of these functions are essential for understanding the rate of change in these systems

### Patterns and Rules

The derivatives of sine and cosine exhibit a cyclical pattern and can be combined with other differentiation rules to solve problems involving trigonometric expressions

## Derivatives of Trigonometric Functions

### Derivatives of Basic Trigonometric Functions

The derivatives of sine, cosine, and tangent are fundamental in calculus and are used to determine the velocity and acceleration of oscillating particles

### Proofs of Derivatives

Sine and Cosine

The derivatives of sine and cosine can be rigorously proven using the definition of the derivative as the limit of the difference quotient

Tangent

The derivative of tangent is derived using the quotient rule, as tangent is the ratio of sine to cosine

### Inverse Trigonometric Functions

The derivatives of arcsine, arccosine, and arctangent are crucial for solving calculus problems involving determining the rate of change of angles given their trigonometric values

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