Cos^(-1) (cos((7\pi )/(4) ))

Question

`cos^(-1) (cos((7\pi )/(4) ))`

Answer 1

For starters,

`\cos\frac{7\pi}4=\cos\left(2\pi-\frac\pi4\right)=\cos\left(-\frac\pi4\right)=\cos\frac\pi4=\frac1{\sqrt2}`

The second equality follows from the fact that
`\cos x` has a period of
`2\pi`, and the equality after that follows from the parity of
`\cos x` (i.e.
`\cos(-x)=\cos x`).

Then

`\cos^(-1)\left(\cos\frac{7\pi}4\right)=\cos^(-1)\left(\cos\frac\pi4\right)=\cos^(-1)\frac1{\sqrt2}=\boxed{\frac\pi4}`