How many solutions does $2 (\sin^{-1}x)^2 -5\sin^{-1}x+2=0$ have?

Question

Number of solutions of the equation
$2 (\sin^{-1}x)^2 -5\sin^{-1}x+2=0 $

Let $t=\sin^{-1}x $

$2 t^2 -5t+2=0$

$(t-2)(2t-1)=0 $

$t=2$ or $t=1/2$

Then

$\sin^{-1}x=2\quad$ or$\quad \sin^{-1}x=1/2$

$x=\sin 2\quad$ or$\quad x=\sin(1/2)$

Did I do something wrong here? I found a solution here => https://www.toppr.com/ask/question/number-of-solutions-of-the-equation-2sin1x25sin1x20/

The solution given on this site says only one solution exists. But I can't understand why $x=\sin 2$ is rejecting. isn't $t$ is an angle and $x$ is a value? Please help.

Answer 1

You have to reject $x=\sin 2$ as $\arcsin x$ is defined on $[-1,1]$ and takes values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ by definition. As $2$ doesn’t belong to the last interval, this solution has to be rejected.

See inverse trigonometric functions.

Answer 2

The (real-valued) arcsine returns a number in $[-\pi/2,\pi/2]$. In particular $\arcsin\sin2$ is not $2$, but $\pi-2$, so it doesn't work.

Answer 3

I disagree with the other responses as well as the official answer, because I consider the normal range of the ArcSin function as irrelevant.

Consider the equation $t^2 = 2.$ The fact that the square root function has a normal range of non-negative values only is irrelevant. The equation $t^2 = 2$ still has two solutions, not one.

Just because the ArcSin function normally has a restricted range does not render the equation ArcSin$(x) = 2$ as gibberish. Here, $x$ can be reasonably interpreted as representing the $\sin(2)$.

Ultimately, it comes down to interpretation. Do you interpret the equation ArcSin$(x) = 2$ to mean that $2$ is the result of the ArcSin function at some value $x$ (which would imply that this equation has no solution), or do you interpret the equation as find $x$ such that $\sin(2) = x.$

Note that the $\sin^{(-1)}$ syntax is being used instead of ArcSin. It is not that unusual, when faced with a non-injective function $f(x)=y$ to interpret $f^{(-1)}(y)$ to represent all of the values of $x$ such that $f(x) = y.$

Very similarly, do you interpret the equation $t^2 = 2$ as find $t$ such that $t^2 = 2$, or do you interpret it as find $t$ such that $t = \sqrt{2}.$