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Pls, can you help me? thx.

\cos(x + (\pi)/(3)) \geqslant ( √(2) )/(2)

User Anubhab
by
8.2k points

1 Answer

1 vote

For
x between
-\pi and
\pi, we have


  • \cos x=\frac{\sqrt2}2=\frac1{\sqrt2} when
    x=\pm\frac\pi4;

  • \cos x=1 for
    x=0; and

  • \cos x=0 for
    x=\pm\frac\pi2


\cos x is continuous over its domain, so the intermediate value theorem tells us that


\cos x\ge\frac{\sqrt2}2

is true for
-\frac\pi4\le x\le\frac\pi4.

For all
x, we take into account that
\cos x is
2\pi-periodic, so the above inequality can be expanded to


-\frac\pi4\le x+2n\pi\le\frac\pi4

where
n is any integer. Equivalently,


-\frac\pi4-2n\pi\le x\le\frac\pi4-2n\pi

To get the corresponding solution set for


\cos\left(x+\frac\pi3\right)\ge\frac{\sqrt2}2

simply replace
x with
x+\frac\pi3:


-\frac\pi4-2n\pi\le x+\frac\pi3\le\frac\pi4-2n\pi


\implies\boxed{-(7\pi)/(12)-2n\pi\le x\le-\frac\pi{12}-2n\pi}

User Sergey Onishchenko
by
8.3k points

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