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

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

Let cos θ = x where, (- 1 ≤ x ≤ 1) then θ = cos$$^{-1}$$ x.

Here θ has infinitely many values.

Let 0 ≤ α ≤ $$\frac{π}{2}$$, where α is positive smallest numerical value and satisfies the equation cos θ = x then the angle α is called the principal value of cos$$^{-1}$$ x.

Again, if the principal value of cos$$^{-1}$$ x is α (0 ≤ α ≤ π) then its general value = 2nπ ± α

Therefore, cos$$^{-1}$$ x = 2nπ ± α, where, 0 ≤ α ≤ π and (- 1 ≤ x ≤ 1).

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

1. Find the General and Principal Values of cos$$^{-1}$$ ½

Solution:

Let x = cos$$^{-1}$$ ½

⇒ cos x = ½

⇒ cos x = cos $$\frac{π}{3}$$

⇒ x = $$\frac{π}{3}$$

⇒ cos$$^{-1}$$ ½ = $$\frac{π}{3}$$

Therefore, principal value of cos$$^{-1}$$ ½ is $$\frac{π}{3}$$ and its general value = 2nπ ± $$\frac{π}{3}$$.

2. Find the General and Principal Values of cos$$^{-1}$$ (-½)

Solution:

Let x = cos$$^{-1}$$ (-½)

⇒ cos x = (-½)

⇒ cos x = - cos $$\frac{π}{3}$$

⇒ cos x = cos (π - $$\frac{π}{3}$$)

⇒ x = $$\frac{2π}{3}$$

⇒ cos$$^{-1}$$ (-½) = $$\frac{2π}{3}$$

Therefore, principal value of cos$$^{-1}$$ (-½) is $$\frac{2π}{3}$$ and its general value = 2nπ ± $$\frac{2π}{3}$$.

Inverse Trigonometric Functions