Question

[tex]in cos(3x)+1, why isn't The "Period" =pi/3?

Shift should be "0", I hope.
What's the range?[/tex]

Answer 1

Let the period be
`p\\eq0`. Then by definition, a
`p`-periodic function
`f(x)` satisfies


`f(x)=f(x+p)`

where here
`f(x)=\cos3x`.


`\cos3x=\cos3(x+p)=\cos(3x+3p)`

We know that
`\cos x` is
`2\pi`-periodic, i.e.


`\cos x=\cos(x+2\pi)`

which reveals that we should also have


`\cos3x=\cos(3x+2\pi)`

and so
`p` is such that
`3p=2\pi\implies p=\frac{2\pi}3`.

In general, the period of
`\cos nx` is
`\frac{2\pi}n`. (The same applies for
`\sin nx`.)

The range can be found by recalling that
`-1\le\cos3x\le1`, which means
`0\le\cos3x+1\le2`.