The Limit of sin(theta)/theta

Definition

The limit of the ratio $\frac{\sin \theta}{\theta}$ as $\theta$ approaches zero is a fundamental trigonometric limit that equals $1$, provided $\theta$ is measured in radians.

Why It Matters

This is the “fundamental constant” of trigonometry. It proves that for very small angles, a curve is nearly a straight line—the essential approximation that allows all of calculus to handle trigonometric functions without collapsing.

Core Concepts

• Fundamental Theorem: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$

  • How to read: “The limit as theta approaches zero of the ratio of sine theta to theta equals one.”
  • Meaning / when to use: Near zero, arc length and vertical rise on the unit circle are nearly equal, so their ratio tends to 1—essential for all trig derivatives.

• Proof Strategy: Uses the Sandwich Theorem by establishing the inequality $\cos \theta < \frac{\sin \theta}{\theta} < 1$ for $\theta \in (-\pi/2, \pi/2), \theta \neq 0$.

  • How to read: “The cosine of theta is less than the ratio of sine theta to theta, which is less than one.”
  • Meaning: Geometric squeeze between a chord and an arc traps the ratio at 1 as $\theta \to 0$.

• Measurement Requirement: This limit holds true only when $\theta$ is in radians; in degrees, the limit is $\frac{\pi}{180}$.

  • How to read: “The quantity pi divided by one hundred eighty.”
  • Meaning: Degree measure scales angles by $\pi/180$; the limit value changes because radians are the natural calculus unit.

Connected Concepts

• Double Integrals over Rectangles
• Precise Definition of a Limit
• Failure of Limits to Exist