Differentiating sin and cos

Differentiating sin and cos

Understanding Sine and Cosine

• The sine and cosine functions are essential expertise for the calculus part of higher mathematics.
• These functions are periodic and relate the angles of a triangle to the dimensions of the triangle.
• Recall that sin(x) and cos(x) are functions that repeat every 2π, also known as their period.

Derivatives of Sine and Cosine

• The derivative of sin(x) is cos(x). This means that the slope of the sin(x) function at any point x is equivalent to cos(x).
• Conversely, the derivative of cos(x) is -sin(x). If you are given a function that is the cosine of x, you can find its derivative by negating the sine of x.
• A key point to remember is that the derivative of sin(x) is positive while the derivative of cos(x) is negative.

Higher Derivatives

• The second derivative of sin(x) is -sin(x), achieved by differentiating cos(x).
• The second derivative of cos(x) is -cos(x), obtained by differentiating -sin(x).
• Interestingly, the second derivative of both these functions return you back to the negative of the original function.

Applications of Derivatives

• These derivatives are used in many fields of study, including physics, engineering, and computer science.
• They allow you to determine rates of change and understand the behaviour of dynamic systems.
• For example, they can help in understanding wave patterns and their properties.