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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