The figure at the right shows a sector of a circle with radius 1. The sector is θ/(2π) of the whole circle, so its area is θ/2.

The area of triangle OAD is AB/2, or sinθ/2. The area of triangle OCD is CD/2, or tanθ/2.

Since triangle OAD lies completely inside the sector, which in turn lies completely inside triangle OCD, we have

This geometric argument applies if 0<θ<π/2. It relies on definitions of arc length and 
area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than
a provable property. For the sine function, we can handle other values. If θ>π/2, then θ>1. But sinθ≤1 (because of the Pythagorean identity), so sinθ<θ. So we have

For negative values of θ we have, by symmetry of the sine function

Hence

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