General and Principal Values of sec\(^{-1}\)  x

How to find the general and principal values of sec\(^{-1}\) x?

Let sec θ = x (| x | ≥ 1 i.e., x ≥ 1 or, x ≤ - 1) then θ = sec - 1x .

Here θ has infinitely many values.

Let 0 ≤ α ≤ π, where α is (α ≠ \(\frac{π}{2}\)) non-negative smallest numerical value of these infinite number of values and satisfies the equation sec θ = x then the angle α is called the principal value of sec\(^{-1}\) x.

Again, if the principal value of sec\(^{-1}\) x is α (0 < α < π) and α ≠ \(\frac{π}{2}\)  then its general value = 2nπ ± α, where, | x | ≥ 1.

Therefore, sec\(^{-1}\) x = 2nπ ± α, where, (0 ≤ α ≤ π), | x | ≥ 1 and α \(\frac{π}{2}\).


Examples to find the general and principal values of arc sec x:

1.Find the General and Principal Values of sec \(^{-1}\) 2.

Solution:

Let x = sec\(^{-1}\) 2

sec x = 2

sec x = sec \(\frac{π}{3}\)

x = \(\frac{π}{3}\)

sec\(^{-1}\) 2 = \(\frac{π}{3}\)

Therefore, principal value of sec\(^{-1}\) 2 is \(\frac{π}{3}\) and its general value = 2nπ ± \(\frac{π}{3}\).

 

2. Find the General and Principal Values of sec \(^{-1}\) (-2).

Solution:

Let x = sec\(^{-1}\) (-2)

sec x = -2

sec x = -sec \(\frac{π}{3}\)

sec x = sec (π  - \(\frac{π}{3}\))

sec x = sec \(\frac{2π}{3}\)

x = \(\frac{2π}{3}\)

sec\(^{-1}\) (-2) = \(\frac{2π}{3}\)

Therefore, principal value of sec\(^{-1}\) (-2) is \(\frac{2π}{3}\) and its general value = 2nπ ± \(\frac{2π}{3}\).






11 and 12 Grade Math

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