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

What are the general and principal Values of sin\(^{-1}\) x?

What is sin\(^{-1}\) ½?

We know that sin (30°) = ½.

⇒ sin\(^{-1}\) (1/2) = 30° or \(\frac{π}{6}\).

Again, sin θ = sin (π - \(\frac{π}{6}\))

⇒ sin θ = sin (\(\frac{5π}{6}\))

⇒ θ = \(\frac{5π}{6}\)or 150°

Again, sin θ = 1/2

⇒ sin θ = sin \(\frac{π}{6}\)

⇒ sin θ = sin (2π + \(\frac{π}{6}\))

⇒ sin θ = sin (\(\frac{13π}{6}\))

⇒ θ = \(\frac{13π}{6}\) or 390°

Therefore, sin (30°) = sin (150°) = sin (390°) and so on, and, sin (30°) = sin (150°) = sin (390°) = ½.

In other ward we can say that,

sin (30° + 360° n) = sin (150° + 360° n) = ½, where, where n = 0, ± 1, ± 2, ± 3, …….

And in general, if sin θ = ½ = sin \(\frac{π}{6}\) then θ = nπ + (- 1)\(^{n}\) \(\frac{π}{6}\), where n = 0 or any integer.

Therefore, if sin θ = 1/2 then θ = sin\(^{-1}\) ½ = \(\frac{π}{6}\) or \(\frac{5π}{6}\) or \(\frac{13π}{6}\)

Therefore in general, sin\(^{-1}\)  (½) = θ = nπ + (-1) \(^{n}\) \(\frac{π}{6}\) and the angle nπ + (- 1)\(^{n}\) \(\frac{π}{6}\) is called the general value of sin\(^{-1}\) ½.

The positive or negative least numerical value of the angle is called the principal value

In this case the \(\frac{π}{6}\) is the least positive angle. Therefore, the principal value of sin\(^{-1}\) ½ is \(\frac{π}{6}\).

Let sin θ = x and - 1 ≤ x ≤ 1

x ⇒ sin {nπ + (- 1)\(^{n}\) θ}, where n = 0, ± 1, ± 2, ± 3, …….

Therefore, sin\(^{-1}\) x = nπ + (- 1)\(^{n}\) θ, where n = 0, ± 1, ± 2, ± 3, …….

For the above equation we can say that sin\(^{-1}\) x may have infinitely many values.

Let – \(\frac{π}{2}\) ≤ α ≤ \(\frac{π}{2}\), where α is positive or negative smallest numerical value and satisfies the equation sin θ = x then the angle α is called the principal value of sin\(^{-1}\) x.

Therefore, the general value of sin\(^{-1}\) x is nπ + (- 1)\(^{n}\) θ, where n = 0, ± 1, ± 2, ± 3, …….

The principal value of sin\(^{-1}\) x is α, where - \(\frac{π}{2}\) ≤ α ≤ \(\frac{π}{2}\) and α satisfies the equation sin θ = x.

For example, principal value of sin\(^{-1}\) (-\(\frac{√3}{2}\)) is -\(\frac{π}{3}\)and its general value is nπ + (- 1)\(^{n}\) ∙ (-\(\frac{π}{3}\)) = nπ - (- 1)\(^{n}\) ∙ \(\frac{π}{3}\). 

Similarly, principal value of sin\(^{-1}\) (\(\frac{√3}{2}\)) is (\(\frac{π}{3}\)) and its general value is nπ + (- 1)\(^{n}\) (\(\frac{π}{3}\)) = nπ - (- 1)\(^{n}\) ∙ \(\frac{π}{6}\). 

● 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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