Question

Find the approximate value of tan⁻¹ (1.001).

Answer 1

Let f(y) = tan^-1y

Differentiating f(y) w.r.t.y, we have

⇒ f'(y) = `1/( 1 + y^2)`

y = 1.001 = x + Δx

Here,
x = 1
Δx = 0.001

Therefore, f(x) = f(1) = tan^-1(1) = `π/4`

Similarly, f'(x) = f'(1) = `1/(1 + 1^2) = 1/2`

Now,
f(y) = f( x + Δx ) = f(x) + Δx. f'(x)    ...[ ∵ Δx <<< x ]

tan^-1y = tan^-1( x + Δx ) = tan^-1x + Δx.`(1/( 1 + x^2))`

∴ tan^-11.001 = tan^-1( 1 + 1.001 ) = tan^-11 + (0.001). tan^-11

⇒ tan^-11.001 = `π/4 + 0.001 (1/2)`

⇒ tan^-11.001 = `π/4 + 0.0005` ≈ 0.7855

Hence the approxiate value of tan^-10.001 will be 0.7855.