\[ \sin^{-1}(\sin \theta)=\theta.\] in the above is it true for both \(\sin x\) and \(\sin^{-1}(x)\) for the values \(n,n\pi,2n\pi,\frac{\pi}{2}\)

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TopNewest\( \sin^{-1}(\sin \theta)=\theta.\) it is true for principal value \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) only

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Yup! In the domain of the principal value, these functions are indeed inverses of each other.

Similarly, \( \sqrt{ x^2 } \neq x \) for all real values of \(x \).

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\(\sin^{-1}(\sin x) = x + 2n\pi\)

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When happens when \( x = 5 \pi \)? See this problem

I disagree with step 1 of your approach. \( \sin 5 \pi = 0 \) but \( \sin^{-1} 0 \neq 5 \pi \).

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Yes it is very interesting. so \(\sin^{-1} (0)\neq 5\pi\). keep as it is \(\sin 5\pi\) without substituting zero what will happen!

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