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Derivatives of Sine and Cosine

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

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's $\pi/2$ "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