Question

Pls, can you help me? thx.

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

Answer 1

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}`