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Evaluate each of the following:sin–1(sin4) |
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Answer» sin–1(sin x) = x Provided x ∈ \([\frac{-\pi}2,\frac{\pi}2]\) ≈ [–1.57,1.57] And in our equation x is 4 which does not lie in the above range. We know sin[π – x] = sin[–x] ∴ sin(π – 4) = sin(–4) Also π–4 belongs in \([\frac{-\pi}2,\frac{\pi}2]\) ∴ sin–1(sin 4) = π – 4 sin-1(sin 3)As we know that sin-1(sin x) = x with x ∈ [-π/2, π/2] which is approximately equal to [-1.57, 1.57] But here x = 3, which does not lie on the above range, So, we know that sin (π – x) = sin (x) Thus, sin (π – 3) = sin (3) also π – 3 ∈ [-π/2, π/2] Sin-1(sin 3) = π – 3 |
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