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Exploring the significance of trigonometric functions in representing periodic motion, this overview delves into the derivatives of sine, cosine, tangent, and their reciprocals. It highlights the use of the Chain Rule for complex differentiation and the importance of avoiding common mistakes. The differentiation of inverse trigonometric functions is also discussed, underscoring their role in calculus and the analysis of periodic phenomena.

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## Periodic Motion

### Definition of Periodic Motion

Periodic motion is characterized by movements that repeat at consistent intervals

### Examples of Periodic Motion

Ocean Tides

Ocean tides are an example of periodic motion

Pendulum Swing

The swinging of a pendulum is another example of periodic motion

### Mathematical Modeling of Periodic Motion

Periodic functions, particularly trigonometric functions, are used to mathematically model periodic motion due to their cyclical nature

## Trigonometric Functions

### Definition of Trigonometric Functions

Trigonometric functions are mathematical functions that repeat their values in a predictable pattern

### Primary Trigonometric Functions

Sine (sin) Function

The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle

Cosine (cos) Function

The cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle

Tangent (tan) Function

The tangent function represents the ratio of the opposite side to the adjacent side in a right triangle

### Reciprocal Trigonometric Functions

Cotangent (cot) Function

The cotangent function is the reciprocal of the tangent function

Secant (sec) Function

The secant function is the reciprocal of the cosine function

Cosecant (csc) Function

The cosecant function is the reciprocal of the sine function

## Differentiation

### Definition of Differentiation

Differentiation is the process of finding the rate at which a function's value changes

### Differentiation of Trigonometric Functions

Derivative of Sine Function

The derivative of the sine function is the cosine function

Derivative of Cosine Function

The derivative of the cosine function is the negative sine function

Derivative of Tangent Function

The derivative of the tangent function is the secant squared function

### Chain Rule in Differentiation

The Chain Rule is used to differentiate composite functions, such as trigonometric functions with nested functions

### Common Errors in Differentiating Trigonometric Functions

Common errors include misapplication of signs and confusion between secant and cosecant functions after differentiation

## Inverse Trigonometric Functions

### Definition of Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle in a right triangle given the ratio of sides

### Derivatives of Inverse Trigonometric Functions

Derivative of Arcsine Function

The derivative of the arcsine function is the reciprocal of the square root of 1 minus the square of the input

Derivative of Arctangent Function

The derivative of the arctangent function is the reciprocal of 1 plus the square of the input

Algorino

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