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
11 and 12 Grade Math
From General and Principal Values of arc sin x to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Mar 02, 24 05:31 PM
Mar 02, 24 04:36 PM
Mar 02, 24 03:32 PM
Mar 01, 24 01:42 PM
Feb 29, 24 05:12 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.