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 ± \(\frac{2π}{3}\).







11 and 12 Grade Math

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