What is the period of the f ( x ) = sin ⁡ x + sin ⁡ 3 x? f ( x ) = sin ⁡ x + sin ⁡ 3 x = 2 3 x + x 2 cos ⁡ x − 3 x 2 = 2 sin ⁡ 2 x cos ⁡ x = 4 sin ⁡ x cos 2 ⁡ x f ( x + T ) = 4 sin ⁡ ( x + T ) cos 2 ⁡ ( x + T ) = 4 sin ⁡ x cos 2 ⁡ xhow can I deduct this I have no idea

Question

What is the period of the   f  (  x  )  =  sin  ⁡  x  +  sin  ⁡  3  x?  f  (  x  )  =  sin  ⁡  x  +  sin  ⁡  3  x  =  2            3      x      +      x        2    cos  ⁡            x      −      3      x        2    =  2  sin  ⁡  2  x  cos  ⁡  x  =  4  sin  ⁡  x      cos    2    ⁡  x    f  (  x  +  T  )  =  4  sin  ⁡  (  x  +  T  )      cos    2    ⁡  (  x  +  T  )  =  4  sin  ⁡  x      cos    2    ⁡  xhow can I deduct this I have no idea

stormsteghj · Accepted Answer

The period of       sin  ⁡  x     is       2  π     and that of       sin  ⁡  3  x     is                 2      π        3      , because       sin  ⁡      (    3                  2        π            3        )    =  sin  ⁡  2  π    .Now given that       f  (  x  )  =  sin  ⁡  3  x  +  sin  ⁡  x    Let  a be a period  f(x) , then by the definition      f  (  x  +  a  )  =  f  (  x  )   . Here       f  (  x  +  a  )  =  sin  ⁡  (  3  x  +  3  a  )  +  sin  ⁡  (  x  +  a  )   From       sin  ⁡  (  3  x  +  3  a  )    , we have       a  =            2      π        3    n  ,        n  n is integer, because its the same as adding period of       sin  ⁡  3  x  ,        n times.Similarly, from       sin  ⁡  (  x  +  a  )    , we have       a  =  2  π  k    is integer. a  is a multiple of                 2      π        3       and       2  π    , so the period is smallest positive multiple of                 2      π        3       and       2  π     which is       2  π    , because       2  π  =  2  π  ∗  1  (  multiple of     2  π  )  ,  2  π  =            2      π        3    ∗  3  (  multiple of               2      π        3    )    . The period of  f(x) ,       2  π  =  2  π  ∗  1  (  multiple of     2  π  )  ,  2  π  =            2      π        3    ∗  3  (  multiple of               2      π        3    )

What is the period of the f ( x ) = sin ⁡ x + sin ⁡<!

Answered question

Answer & Explanation

New Questions in Precalculus