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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