Derivatives of Sine and Cosine

6 min read

Why These Two Derivatives Matter

Every oscillating system in nature (a swinging pendulum, a vibrating guitar string, an alternating current) is described by sine and cosine functions. To understand how fast these systems change at any moment, we need their derivatives.

Here’s the beautiful surprise: the derivative of sine is cosine, and the derivative of cosine is negative sine. These two functions are intertwined in a cycle that repeats every four derivatives. This elegant pattern makes trigonometric derivatives some of the most memorable in all of calculus.

Prerequisite Map

Quick Reference

Key Formulas

$$\boxed{\frac{d}{dx}(\sin x) = \cos x}$$

$$\boxed{\frac{d}{dx}(\cos x) = -\sin x}$$

Critical reminder: These formulas are only valid when $x$ is measured in radians. If you work in degrees, you’ll get an extra factor of $\pi/180$.

The Two Essential Limits

Before we can prove these formulas, we need two special limits:

$$\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \qquad \lim_{\theta \to 0} \frac{\cos \theta - 1}{\theta} = 0$$

Why $\frac{\sin \theta}{\theta} \to 1$?

For small angles (in radians), the sine of the angle is approximately equal to the angle itself:

        Arc length = θ (on unit circle)
              ↗
             /
            /|
           / |  Height = sin θ
          /  |
         /θ__|
        O

For small θ: sin θ ≈ θ

The Squeeze Theorem makes this rigorous: $\cos \theta < \frac{\sin \theta}{\theta} < 1$ for small positive $\theta$, and both bounds approach 1.

Proof: Derivative of Sine

Starting from the definition:

$$\frac{d}{dx}(\sin x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$

Using the angle addition formula $\sin(x+h) = \sin x \cos h + \cos x \sin h$:

$$= \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h}$$

$$= \lim_{h \to 0} \left( \sin x \cdot \frac{\cos h - 1}{h} + \cos x \cdot \frac{\sin h}{h} \right)$$

$$= \sin x \cdot 0 + \cos x \cdot 1 = \cos x$$

The Cyclic Pattern

The derivatives of sine and cosine repeat in a cycle of four:

Pattern: Differentiating four times returns you to the original function.

Finding the $n$th derivative: Divide $n$ by 4 and look at the remainder.

• Remainder 0: back to original
• Remainder 1: first derivative
• Remainder 2: second derivative
• Remainder 3: third derivative

Physical Interpretation

For simple harmonic motion with position $s(t) = A\sin(t)$:

• Position: $s = A\sin t$
• Velocity: $v = \frac{ds}{dt} = A\cos t$
• Acceleration: $a = \frac{dv}{dt} = -A\sin t$

Notice: $a = -s$. The acceleration is proportional to position but opposite in sign. This is the defining property of simple harmonic motion.

Position:      ∿∿∿∿∿  (sine wave)
                 ↓ derivative
Velocity:      ∿∿∿∿∿  (cosine = shifted sine)
                 ↓ derivative
Acceleration:  ∿∿∿∿∿  (negative sine)

When position is at maximum, velocity is zero. When position crosses zero, velocity is at maximum.

Practice Problems

CCI-Style Conceptual Questions

     1 |     ∩
       |   /   \
     0 |--/-----\-----/--
       |         \   /
    -1 |          ∪
       0   π/2   π  3π/2  2π

Common Misconceptions

Mastery Checklist

• I can state the derivatives of $\sin x$ and $\cos x$ from memory
• I understand why the derivative of $\cos x$ has a negative sign
• I can differentiate linear combinations like $a\sin x + b\cos x$
• I can apply the product rule with sine and cosine
• I can find the $n$th derivative of $\sin x$ or $\cos x$ using the cyclic pattern
• I can find tangent lines to curves involving sine and cosine
• I know that these formulas require $x$ to be in radians

Mental Model

The Phase Shift Connection:

Think of $\cos x$ as a sine wave shifted left by $\pi/2$:

$$\cos x = \sin\left(x + \frac{\pi}{2}\right)$$

When you differentiate sine, you get cosine, a function that is $\pi/2$ ahead in the wave cycle. Differentiating again shifts another $\pi/2$, giving $-\sin x$ (which is $\pi$ ahead, i.e., the negative).

Four shifts of $\pi/2$ brings you full circle: $4 \times \frac{\pi}{2} = 2\pi$, which is one complete period.

Connections

Looking back:

• The derivative definition is the foundation for proving these formulas
• Limit laws let us split the proof into manageable pieces

Looking ahead:

• Other trig derivatives builds on these using the quotient rule
• The chain rule lets us differentiate $\sin(3x)$, $\cos(x^2)$, etc.

Real-world connections:

• In physics, these derivatives describe velocity and acceleration of oscillating systems
• In electrical engineering, they model alternating current circuits

Last updated: 2026-01-22