3 arcsin(x) = arcsin(3x - 4x\(^{3}\))

We will learn how to prove the property of the inverse trigonometric function 3 arcsin(x) = arcsin(3x - 4x\(^{3}\)) or, 3 sin\(^{-1}\) x = sin\(^{-1}\) (3x - 4x\(^{3}\))

Proof:  

Let, sin\(^{-1}\) x = θ      

Therefore, sin θ = x

Now we know that, sin 3θ = 3 sin θ - 4 sin\(^{3}\) θ

⇒ sin 3θ = 3x - 4x\(^{3}\)

Therefore, 3θ = sin\(^{-1}\) (3x - 4x\(^{3}\))

⇒ 3 sin\(^{-1}\) x = sin\(^{-1}\) (3x - 4x\(^{3}\))

or, 3 arcsin(x) = arcsin(3x - 4x\(^{3}\))           Proved

11 and 12 Grade Math

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