Question

Solve the following : Find the approximate value of cos^–1 (0.51), given π = 3.1416, `(2)/sqrt(3)` = 1.1547.

Answer 1

Let f(x) = cos^–1 x.

Then f'(x) = `d/dx(cos^-1 x) = (-1)/sqrt(1 - x^2)`

Take a = 0.5 and h = 0.01

Then f(a) = f(0.5)

= cos^–1 (0.5)

= `cos^-1(cos  pi/3)`

= `pi/(3)`
and
f'(a) = f'(0.5)

= `-(1)/sqrt(1 - (1/2)^2`

= `-(2)/sqrt(3)`
= – 1.1547
The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ cos^–1 (0.51) = f(0.51)
= f(0.5 + 0.01)
≑ f(0.5) + (0.01)f'(0.5)

≑ `pi/(3) + 0.01 xx (-1.1547)`

≑ `(3.1416)/(3) - 0.011547`

≑ 1.0472 - 0.01157 = 1.035653
∴ cos^–1 (0.51) ≑ 1.035653.