Question

Sin theta =
${x}^{2} + \frac{1}{ {x}^{2} }$
is possible or impossible for any real x ??​

Answer 1

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• Check if sin ( theta ) = x² + 1/x² possible

• [ x belongs to set of real number ]

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• We know, range of sin theta = [ -1, 1 ]

• So that means if the minimum value of the identity is in [ -1, 1 ] , theta will exist for real value of x

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◍ let's start with checking the minimum value of the given identity

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$\dashrightarrow \: \sf f(x)_{min} = {x}^{2} + \frac{1}{ {x}^{2} }$

$\dashrightarrow \: \sf f(x)_{min} = {\bigg({x}^{} - \frac{1}{ {x}^{} } \bigg)}^{2} + 2$

◍ Now,

$\dashrightarrow \sf{\bigg({x}^{} - \frac{1}{ {x}^{} } \bigg)}^{2} \geqslant 0$

$\bf \therefore \: f(x)_{min} = 2$

◍ But,

• Range of sin thera [ -1 , 1 ]

◆ ᴡᴇ ᴄᴀɴ ᴄᴏɴᴄʟᴜᴅᴇ ᴛʜᴇʀᴇ's ɴᴏ sᴏʟᴜᴛɪᴏɴs ғᴏʀ ʀᴇᴀʟ x

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