Medium

Derivatives of Sine and Cosine

Simple Explanation

The derivative of sine is cosine, and the derivative of cosine is negative sine: d/dx[sinx]=cosx and d/dx[cosx]=−sinx.

Why Do We Need It?

These two derivatives are the foundation for differentiating any expression involving trigonometric functions, including all the other trig derivatives in this chapter.

See It

The tangent line to y=sin(x) at the origin, with slope 1

Formula

Derivatives of Sine and Cosine

d/dx[sin x] = cos x, d/dx[cos x] = −sin x

The derivative of sine is cosine; the derivative of cosine is negative sine.

x: — the input angle, in radians

When to use it: Whenever differentiating an expression involving sine or cosine.

Worked Example

Differentiate an expression with sine and cosine

Find the derivative of f(x) = 3sinx − 2cosx.

Why Does This Work?

Near x=0, the sine curve is essentially indistinguishable from the straight line y=x (its own tangent line there) — this matches d/dx[sinx]=cosx, since cos(0)=1, giving exactly slope 1 at that point, consistent with the graph shown.

Real-Life Example

Finding the velocity of an oscillating object

An object's position over time follows a sine curve (simple harmonic motion), and its velocity is needed at a given instant.

Velocity is the derivative of position — differentiating the sine position function using d/dx[sinx]=cosx gives the velocity function directly.

Practice

Find f'(0) for f(x)=sinx+cosx.

Medium

Common mistake

Forgetting the negative sign in the derivative of cosine — d/dx[cosx]=−sinx, not +sinx.

Quick Review

d/dx[sinx] = cosx.
d/dx[cosx] = −sinx (note the negative sign).
These are the foundation for all other trig derivatives.