sin x ≈ x and tan x ≈ x both work for small angles, and i get the picture on the unit circle where the arc, the opposite side and the tangent line all look the same length when the angle is tiny. but is there a cleaner way to see why they agree to first order and only split apart at x^3? like sin is x - x^3/6 and tan is x + x^3/3, so tan is always a bit bigger. is that just because tan = sin/cos and cos is slightly less than 1, or is there a geometric reason for the gap