PROOF 131

Prove.

The limit of sin x divided by x is 1

On the unit circle.

What you may use

A1X ⊆ Y ⟹ [X] ≤ [Y]
Monotonicity of area
A2area of a triangle = ½bh
Area of a triangle
A3area of a sector = ½r²x
Area of a sector (radians, Proof 50)
A4△OAP ⊆ S ⊆ △OAQ
Inclusion relations (0 < x < π/2)
A50<x<π/2 ⟹ 0<sin x, cos x
Coordinate signs in the first quadrant
A6sin(−x) = −sin x
sin is odd
A7cos(−x) = cos x
cos is even
A8cos x ≤ 1
The x-coordinate on the unit circle
A91 − cos θ = 2 sin²(θ/2)
Half-angle formula (from the addition theorem, Proof 34)
A100<c ⟹ (a≤b ⟺ ac≤bc)
Order axiom (positive multiplier)
A11a ≤ b ≤ c, one strict ⟹ a<c
Transitivity of order
A120 < a ⟹ 0 < 1/a
Sign of the reciprocal
A130 ≤ a ≤ b ⟹ a² ≤ b²
Squares and order
A140 < a ⟹ 0 < a/2 < a
Half of a positive number
A15min(a,b,c) ≤ a, b, c
The smallest (one of the three)
A160 < 1, 0 < 2, 0 < π/2
Positive numbers
A17|a| = a (a≥0), −a (a<0)
Meaning of absolute value
A18g ≤ f ≤ h, g,h→L ⟹ f→L
Squeeze theorem
A19lim c = c
Limit of a constant function
A20associativity, commutativity, distributivity, signs
Arithmetic of the reals
A21a ≠ 0 ⟹ 0 < a or a < 0
Trichotomy of order
Ea = b ⟹ apply the same operation to both sides
Properties of equality

Nothing outside this list may be cited.

Definition D1

The circle of radius 1 centered at the origin O is called the unit circle, and A = (1, 0). Let P = (cos x, sin x) be the point where the ray making angle x counterclockwise from the positive x-axis meets the unit circle, and let the sector S be the region bounded by the two radii OA, OP and the arc. Define tan x = sin x / cos x.

Proof.

Related labs

Limit lab →Trigonometric Function lab →