The-maximum-value-of-f-x-sin-2-2pi-3-x-sin-2-2pi-3-x-is-

Question Number 130054 by EDWIN88 last updated on 22/Jan/21

The maximum value of f (x) = \sin^{2} (\frac{2 π}{3} + x) + \sin^{2} (\frac{2 π}{3} - x) is ?

Answered by liberty last updated on 22/Jan/21

Commented by MJS_new last updated on 22/Jan/21

0 ⩽ \sin^{2} α ⩽ 1

\Rightarrow 0 ⩽ \sin^{2} α + \sin^{2} β ⩽ 2

answer has to be within this interval

Commented by MJS_new last updated on 22/Jan/21

error in line 3, it musr be - \cos 2 x

Commented by liberty last updated on 22/Jan/21

thank you sir

Answered by MJS_new last updated on 22/Jan/21

\sin^{2} (a - x) + \sin^{2} (a + x) =

= {(\sin a \cos x - \cos a \sin x)}^{2} + {(\sin a \cos x + \cos a \sin x)}^{2} =

= 2 (\cos^{2} a \sin^{2} x + {2 \sin}^{2} a \cos^{2} x) =

= 2 ((1 - \sin^{2} a) \sin^{2} x + {2 \sin}^{2} a (1 - \sin^{2} x)) =

= 2 (\sin^{2} a + (1 - {2 \sin}^{2} a) \sin^{2} x) =

[a = \frac{2 π}{3}]

= \frac{3}{2} - \sin^{2} x

\frac{1}{2} ⩽ \frac{3}{2} - \sin^{2} x ⩽ \frac{3}{2}

Commented by liberty last updated on 22/Jan/21

haha . . what wrong my solution

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