General and Principal Values of csc\(^{-1}\)  x

How to find the general and principal values of ccs\(^{-1}\) x?

Let csc θ = x (| x |≥ 1 i.e., x ≥ 1 or, x ≤ - 1) then θ = csc\(^{-1}\) x .

Here θ has infinitely many values.

Let – \(\frac{π}{2}\) ≤ α ≤ \(\frac{π}{2}\), where α is non-zero(α ≠ 0) positive or negative smallest numerical value of these infinite number of values and satisfies the equation csc θ = x then the angle α is called the principal value of csc\(^{-1}\) x.

Again, if the principal value of csc\(^{-1}\) x is α (– \(\frac{π}{2}\) < α < \(\frac{π}{2}\)) and α ≠ 0 then its general value = nπ + (- 1) n α, where, | x | ≥ 1.

Therefore, tan\(^{-1}\) x = nπ + α, where, (– \(\frac{π}{2}\) < α < \(\frac{π}{2}\)), | x | ≥ 1 and (- ∞ < x < ∞).

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

1. Find the General and Principal Values of csc \(^{-1}\) (√2).

Solution:

Let x = csc\(^{-1}\) (√2)

⇒ csc x = √2

⇒ csc x = csc \(\frac{π}{4}\)

⇒ x = \(\frac{π}{4}\)

⇒ csc\(^{-1}\) (√2) = \(\frac{π}{4}\)

Therefore, principal value of csc\(^{-1}\) (√2) is \(\frac{π}{4}\) and its general value = nπ + (- 1)\(^{n}\) ∙ \(\frac{π}{4}\).

2. Find the General and Principal Values of csc \(^{-1}\) (-√2).

Solution:

Let x = csc\(^{-1}\) (-√2)

⇒ csc x = -√2

⇒ csc x = csc (-\(\frac{π}{4}\))

⇒ x = -\(\frac{π}{4}\)

⇒ csc\(^{-1}\) (-√2) = -\(\frac{π}{4}\)

Therefore, principal value of csc\(^{-1}\) (-√2) is -\(\frac{π}{4}\) and its general value = nπ + (- 1)\(^{n}\) ∙ (-\(\frac{π}{4}\)) = nπ - (- 1)\(^{n}\) ∙ \(\frac{π}{4}\).

● 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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