Library
Mathematics

The Geometric Intuition of the Derivative of Sine

This lesson explores the derivative of sin(x) by visualizing the unit circle, showing how a change in the angle translates into a vertical change along the curve.

Scene 1 of 4
Visualizing Sine on the Circle
y=sin⁡(θ)y = \sin(\theta)
Imagine a particle moving around the unit circle. The height of this particle at any angle theta is defined as sin(theta).
Step-by-step solver
1

Definition

Start with the limit definition of the derivative for the sine function.

f′(x)=lim⁡h→0sin⁡(x+h)−sin⁡(x)hf'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}
2

Angle Sum Identity

Expand sin(x+h) using the sum formula sin(A+B) = sin(A)cos(B) + cos(A)sin(B).

sin⁡(x)cos⁡(h)+cos⁡(x)sin⁡(h)−sin⁡(x)h\frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h}
3

Regrouping terms

Group terms by sin(x) and cos(x) to isolate the limits.

sin⁡(x)(cos⁡(h)−1h)+cos⁡(x)(sin⁡(h)h)\sin(x) \left( \frac{\cos(h)-1}{h} \right) + \cos(x) \left( \frac{\sin(h)}{h} \right)
4

Evaluating limits

Apply the fundamental trigonometric limits where sin(h)/h goes to 1 and (cos(h)-1)/h goes to 0.

sin⁡(x)⋅0+cos⁡(x)⋅1=cos⁡(x)\sin(x) \cdot 0 + \cos(x) \cdot 1 = \cos(x)

Original question

Explain why the derivative of sin(x) is cos(x), intuitively.

Follow-up chat

Ask me anything about this lesson — I'll answer using what we just covered.