Find the range of $sinx(sinx+cosx)$ ?

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Answer 1 · 2017-04-01T17:53:21

Range of $sinx(sinx+cosx)$ is $[1/2-1/sqrt2,1/2+1/sqrt2]$

$sinx(sinx+cosx)$

= $sinxsqrt2(sinx1/sqrt2+cosx1/sqrt2)$

= $sqrt2sinx(sinxcos45^@+cosxsin45^@)$

= $sqrt2sinxsin(x+45^@)$

= $sqrt2/2(2sinxsin(x+45^@))$

= $1/sqrt2[cos(x-(x+45^@)-cos(x+(x+45^@)]$

= $1/sqrt2[cos(-45^@)-cos(2x+45^@)]$

= $1/sqrt2[1/sqrt2-cos(2x+45^@)]$

= $1/2-1/sqrt2cos(2x+45^@)$

As range of $cos(2x+45^@)$ is $[-1,1]$

range of $sinx(sinx+cosx)$ or $1/2-1/sqrt2cos(2x+45^@)$ is

$[1/2-1/sqrt2,1/2+1/sqrt2]$

Find the range of sinx(sinx+cosx)sinx(sinx+cosx)?