Trig Limit & Derivative Explorer: The MIT 18.01 Geometric Lab

Arc θ sin(θ) 1 - cos(θ) tan(θ)

Zoom:

Scrub Angle θ: 0.250 rad (14.3°) Drag point on circle or scrub slider

0.00 Rad (Limit Boundary) 0.75 Rad 1.50 Rad (~86°)

Sector Area Squeeze Triangles Chord vs Arc Tangent Line

Fundamental Limits Telemetry MIT 18.01 Core

First Core Limit: lim_θ→0 sin(θ) / θ = 1

sin(θ) / θ ratio: 0.989616

Vertical leg sin(θ) converges directly into circular arc length θ as θ shrinks.

Second Core Limit: lim_θ→0 (1 - cos(θ)) / θ = 0

(1 - cos(θ)) / θ ratio: 0.123709

Radial deficit (1 - cos θ) shrinks quadratically (~θ²/2), vanishing faster than linear arc θ.

Geometric Area Squeeze Bounds Area Hierarchy

½ sin(θ) ≤ ½ θ ≤ ½ tan(θ)

Inner Tri: 0.1237 ≤ Sector: 0.1250 ≤ Outer Tri: 0.1283

Dividing by ½ sin(θ) gives: 1 ≤ θ/sin(θ) ≤ 1/cos(θ).
Taking reciprocals yields the squeeze: cos(θ) ≤ sin(θ)/θ ≤ 1.

Lower Bound cos(θ): 0.968912 Upper Bound: 1.000000

Trig Derivative Step Workbench

Step 1: Difference Quotient Definition Limit Setup

f'(x) = lim_h→0 [sin(x + h) - sin(x)] / h

Step 2: Angle Sum Expansion Identity

= lim_h→0 [sin(x)cos(h) + cos(x)sin(h) - sin(x)] / h

Step 3: Group by Fundamental Limits MIT 18.01 Split

= sin(x)·[lim_h→0 (cos(h)-1)/h] + cos(x)·[lim_h→0 sin(h)/h]

Step 4: Substitute Known Limits (0 and 1) Evaluated

= sin(x)·(0) + cos(x)·(1) = cos(x)

"To differentiate sine and cosine, you don't need a new rule. You need two limits." — Prof. David Jerison

Two limits behind sine and cosine derivatives

Your browser does not support video.

Read the explanation

For positive angles below pi over two, sine theta is less than theta, which is less than tangent theta. Rearranging gives cosine theta less than sine theta divided by theta, less than one. At the source squeeze preset zero point four radians, the lower bound is about zero point nine two one one and the ratio about zero point nine seven three five. As the angle shrinks both bounds approach one. Degrees cannot be substituted into the same formula without conversion. The derivative preset uses zero point zero eight radians. Its sine over angle is about zero point nine nine eight nine three four, while cosine is about zero point nine nine six eight zero two. One minus cosine divided by the angle is about zero point zero three nine nine seven nine. The second ratio approaches zero as the angle approaches zero, unlike sine over angle which approaches one. The animated bars show the different magnitudes directly. Expanding sine of x plus h splits the difference quotient into sine x times cosine h minus one over h, plus cosine x times sine h over h. The first limit is zero, and the second is one, leaving cosine x. At x pi over three that derivative is one half. Expanding cosine instead gives minus sine x, which at pi over three is minus square root three over two. The original selector displays these algebraic alternatives; finite numerical ratios illustrate rather than prove the limiting argument.