# 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 ± $$\frac{2π}{3}$$.

Inverse Trigonometric Functions