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

• General and Principal Values of sin\(^{-1}\) x
• General and Principal Values of cos\(^{-1}\) x
• General and Principal Values of tan\(^{-1}\) x
• General and Principal Values of csc\(^{-1}\) x
• General and Principal Values of sec\(^{-1}\) x
• General and Principal Values of cot\(^{-1}\) x
• Principal Values of Inverse Trigonometric Functions
• General Values of Inverse Trigonometric Functions
  
• arcsin(x) + arccos(x) = \(\frac{π}{2}\)
• arctan(x) + arccot(x) = \(\frac{π}{2}\)
• arctan(x) + arctan(y) = arctan(\(\frac{x + y}{1 - xy}\))
• arctan(x) - arctan(y) = arctan(\(\frac{x - y}{1 + xy}\))
• arctan(x) + arctan(y) + arctan(z)= arctan\(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)
• arccot(x) + arccot(y) = arccot(\(\frac{xy - 1}{y + x}\))
• arccot(x) - arccot(y) = arccot(\(\frac{xy + 1}{y - x}\))
• arcsin(x) + arcsin(y) = arcsin(x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\))
• arcsin (x) - arcsin(y) = arcsin (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\))
• arccos (x) + arccos(y) = arccos(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
• arccos(x) - arccos(y) = arccos(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
• 2 arcsin(x) = arcsin(2x\(\sqrt{1 - x^{2}}\)) 
• 2 arccos(x) = arccos(2x\(^{2}\) - 1)
• 2 arctan(x) = arctan(\(\frac{2x}{1 - x^{2}}\)) = arcsin(\(\frac{2x}{1 + x^{2}}\)) = arccos(\(\frac{1 - x^{2}}{1 + x^{2}}\))
• 3 arcsin(x) = arcsin(3x - 4x\(^{3}\))
• 3 arccos(x) = arccos(4x\(^{3}\) - 3x)
• 3 arctan(x) = arctan(\(\frac{3x - x^{3}}{1 - 3 x^{2}}\))
• Inverse Trigonometric Function Formula
• Principal Values of Inverse Trigonometric Functions
• Problems on Inverse Trigonometric Function

11 and 12 Grade Math

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