Question

The number of values of x satisfying the equation x=sin^-1 x​

Answer 1

SOLUTION

TO DETERMINE

The number of values of x satisfying the equation

$\displaystyle \sf{x = { \sin}^{ - 1} x}$

EVALUATION

Here the given equation is

$\displaystyle \sf{x = { \sin}^{ - 1} x}$

We now solve for x as below

$\displaystyle \sf{ \implies \sin x = x}$

$\displaystyle \sf{ \implies x - \frac{ {x}^{3} }{3!} + \frac{ {x}^{5} }{5!} - .. \: .. = x}$

$\displaystyle \sf{ \implies - \frac{ {x}^{3} }{3!} + \frac{ {x}^{5} }{5!} - .. \: .. = 0}$

$\displaystyle \sf{ \implies {x}^{3} \bigg( - \frac{ 1 }{3!} + \frac{ {x}^{2} }{5!} - .. \: .. \bigg)= 0}$

$\displaystyle \sf{ \implies {x}^{3} = 0 \: \: \: \: \: \:\bigg( \because - \frac{ 1 }{3!} + \frac{ {x}^{2} }{5!} - .. \: .. \ne \: 0 \bigg)}$

$\displaystyle \sf{ \implies x = 0}$

Hence the required solution is is x = 0

FINAL ANSWER

The number of values of x satisfying the equation is 1 which is x = 0

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