Evaluate each of the following:\(sin^{-1}(sin\frac{13\pi}7)\)

1.	Evaluate each of the following:\(sin^{-1}(sin\frac{13\pi}7)\)
Answer» We can write \((sin\frac{13\pi}7)\) as \(sin(2\pi-\frac\pi{7})\) As we know sin(2π –θ) = sin(–θ ) So, \(sin(2\pi-\frac\pi{7})\) can be written as \(sin(\frac\pi{7})\) ∴ The equation becomes sin^–1\((sin\frac{\pi}7)\) As sin^-1(sin x) = x Provided x ∈ \([\frac{-\pi}2,\frac{\pi}2]\) ∴ we can write \(sin^{-1}(sin\frac{\pi}7)=\frac{\pi}7\)

Answer»

We can write \((sin\frac{13\pi}7)\) as \(sin(2\pi-\frac\pi{7})\)

As we know sin(2π –θ) = sin(–θ )

So, \(sin(2\pi-\frac\pi{7})\) can be written as \(sin(\frac\pi{7})\)

∴ The equation becomes sin^–1\((sin\frac{\pi}7)\)

As sin^-1(sin x) = x

Provided x ∈ \([\frac{-\pi}2,\frac{\pi}2]\)

∴ we can write \(sin^{-1}(sin\frac{\pi}7)=\frac{\pi}7\)

Discussion