period of function ; cos cube 3x+ tan 4x

Question

period of function

Answer 1

if a periodic function f has a period of 12 then does f(12)=f(36)

Answer 2

The period is 2π.

cos³(3x)+tan(4x)=1 when x=0; also =1 when x=2π.

When x=π/3, cos³(3x)=-1, tan(4x)=√3; x=7π/3, cos³(3x)=-1, tan(4x)=√3.

More generally:

cos(3(x+2πn))=cos(3x+6πn)=cos(3x); tan(4(x+2πn))=tan(4x+8πn)=tan(4x).

It's because integer multiples of 2π have the same effect as just adding 2π.

For non-integer multiples (for example, ⅓, ¼, the period is 12π) the period is the LCM of 2 and the fractions multiplied by π. For example, 3/7 and 2/3 would have a period of 42π, not 21π. If you graph the function, the period is the x (horizontal) displacement between exact replications of the shape of the graph. When there is only one trig function involved then the period is given by 2π/B where B is the x (the variable's) coefficient.