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