Question

Find the approximate values of : tan^–1(0.999)

Answer 1

Let f(x) = tan^–1x

Then f'(x) = `d/dx(tan^-1x) = (1)/(1 + x^2)`

Take a = 1 and h = – 0.001

Then f(a) = f(1) = tan^–11 = `pi/(4)`

and f'(a) = f'(1) = `(1)/(1 + 1^2) = (1)/(2)`
The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ tan^–1 (0.999)
= f(0.999)
= f(1 – 0.001)
≑ f(1) – (0.001).f'(1)

≑ `pi/(4) - 0.001 xx (1)/(2)`

= `pi/(4) - 0.0005`

∴ tan^–1 (0999) ≑ `pi/(4) - 0.0005`.
Remark: The answer can also be given as :

tan^–1 (0.999) ≑ `(3.1416)/(4) - 0.0005`

≑ 0.7854 – 0.0005
= 0.7849.