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
From General and Principal Values of arc sec x to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Oct 10, 24 05:15 PM
Oct 10, 24 10:06 AM
Oct 10, 24 03:19 AM
Oct 09, 24 05:16 PM
Oct 08, 24 10:53 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.