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 ± \(\frac{2π}{3}\).
● Inverse Trigonometric Functions
11 and 12 Grade Math
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