Arc θ
sin(θ)
1 - cos(θ)
tan(θ)
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0.00 Rad (Limit Boundary)
0.75 Rad
1.50 Rad (~86°)
Fundamental Limits Telemetry MIT 18.01 Core
First Core Limit:
limθ→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:
limθ→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.
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) = limh→0 [sin(x + h) - sin(x)] / h
Step 2: Angle Sum Expansion
Identity
= limh→0 [sin(x)cos(h) + cos(x)sin(h) - sin(x)] / h
Step 3: Group by Fundamental Limits
MIT 18.01 Split
= sin(x)·[limh→0 (cos(h)-1)/h] + cos(x)·[limh→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