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 $n,n\pi,2n\pi,\frac{\pi}{2}$

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

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

Calvin Lin Staff - 5 years, 9 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!

Siva Raj - 5 years, 9 months ago

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

Janardhanan Sivaramakrishnan - 5 years, 9 months ago

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

Siva Raj - 5 years, 9 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 $x$ .

Calvin Lin Staff - 5 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$