Question

By considering standard series of sin theta and cos theta, show that tan theta is approx = theta when theta is small such that theta^3 and higher powers of theta are negligible.

Answer 1

since

sinx = x - x^3/3! + x^5/5! - ...
cosx = 1 - x^2/2! + x^4/4! - ...
when x is small, higher powers of x become negligible.
So
sinx ≈ x
cosx ≈ 1
and thus tanx = sinx/cosx ≈ x
when x is small.