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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 ≠ π2) 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 ≠ π2 then its general value = 2nπ ± α, where, | x | ≥ 1.
Therefore, sec−1 x = 2nπ ± α, where, (0 ≤ α ≤ π), | x | ≥ 1 and α ≠ π2.
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 π3
⇒ x = π3
⇒ sec−1 2 = π3
Therefore, principal value of sec−1 2 is π3 and its general value = 2nπ ± π3.
2. Find the General and Principal Values of sec −1 (-2).
Solution:
Let x = sec−1 (-2)
⇒ sec x = -2
⇒ sec x = -sec π3
⇒ sec x = sec (π - π3)
⇒ sec x = sec 2π3
⇒ x = 2π3
⇒ sec−1 (-2) = 2π3
Therefore, principal value of sec−1 (-2) is 2π3 and its general value = 2nπ ± 2π3.
● Inverse Trigonometric Functions
11 and 12 Grade Math
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