Derivatives of Trigonometric Functions

Introduction

Trigonometric functions such as sine and cosine are fundamental in modeling periodic phenomena in physics, engineering, and other sciences. Their derivatives are essential for analyzing oscillatory motion, waves, and other periodic behaviors. This section covers the differentiation of the six basic trigonometric functions, with a focus on sine and cosine, and explores their applications.

Derivative of the Sine Function

The derivative of the sine function can be derived using the definition of the derivative and the angle-sum identity for sine:

• Definition:

• Derivative:

• Using the angle-sum identity:

• Simplifying and applying standard limits:

• Result:

Example: Differentiate

Derivative of the Cosine Function

The derivative of the cosine function is found similarly, using the cosine angle-sum identity:

• Definition:

• Derivative:

• Using the angle-sum identity:

• Simplifying and applying standard limits:

• Result:

Example: Differentiate

Derivative of the Tangent Function

The derivative of the tangent function is derived using the quotient rule and the derivatives of sine and cosine:

• Definition:

• Derivative:

Example: Differentiate

Derivatives of the Other Basic Trigonometric Functions

The derivatives of the remaining trigonometric functions are as follows:

Example: Prove Start with and apply the quotient rule.

Applications: Simple Harmonic Motion

Trigonometric functions are widely used to model oscillatory motion, such as a mass on a spring. For example, the position of an object attached to a spring can be given by .

• Velocity:

• Acceleration:

• The object oscillates between cm and cm, with a period of .

Higher-Order Derivatives of Trigonometric Functions

The derivatives of sine and cosine repeat in cycles of four:

Example: The 27th derivative of is .

Standard Trigonometric Limits

Some important limits involving trigonometric functions are:

Example:

Summary Table: Derivatives of Trigonometric Functions

Additional Examples

• Differentiate :

• Differentiate :

• Differentiate :

Application: Oscillating Mass on a Spring

Given (position in cm, in seconds):

• Velocity:

• Acceleration:

• At : cm, cm/s (moving left), cm/s