# General and Principal Values of cot$$^{-1}$$  x

How to find the general and principal values of cot$$^{-1}$$ x?

Let cot θ = x (- ∞ < x < ∞) then θ = cot$$^{-1}$$ x.

Here θ has infinitely many values.

Let – $$\frac{π}{2}$$ ≤ α ≤ $$\frac{π}{2}$$, where α is positive or negative smallest numerical value of these infinite number of values and satisfies the equation cot θ = x then the angle α is called the principal value of cot$$^{-1}$$ x.

Again, if the principal value of cot$$^{-1}$$ x is α (α ≠ 0, – π/2 ≤ α ≤ π/2) then its general value = nπ + α.

Therefore, cot$$^{-1}$$ x = nπ + α, where, (α ≠ 0, – π/2 ≤ α ≤ π/2) and ( - ∞ < x < ∞ ).

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

1. Find the General and Principal Values of cot$$^{-1}$$ √3

Solution:

Let x = cot$$^{-1}$$ √3

⇒ cot x = √3

⇒ cot x = tan (π/6)

⇒ x = π/6

⇒ cot$$^{-1}$$ √3 = π/6

Therefore, principal value of cot$$^{-1}$$ √3 is π/6 and its general value = nπ + π/6.

2. Find the General and Principal Values of cot$$^{-1}$$ (- √3)

Solution:

Let x = cot$$^{-1}$$ (-√3)

⇒ cot x = -√3

⇒ cot x = cot (-π/6)

⇒ x = -π/6

⇒ cot$$^{-1}$$ (-√3) = -π/6

Therefore, principal value of cot$$^{-1}$$ (-√3) is -π/6 and its general value = nπ - π/6.

Inverse Trigonometric Functions

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