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)\(^{n}\) ∙ \(\frac{π}{6}\). 

11 and 12 Grade Math

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