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True or false

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

Note by Siva Raj
2 years, 4 months ago

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

- 2 years, 4 months ago

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 $$x$$.

Staff - 2 years, 4 months ago

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

- 2 years, 4 months ago

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

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

Staff - 2 years, 4 months ago

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!

- 2 years, 4 months ago