Show that a function of the form x ( t ) = K 1 cos ⁡ β t + K 2 sin ⁡ β tCan be written as x ( t ) = K cos ⁡ ( B t − ϕ )Where K = K 1 2 + K 2 2 I know that linear systems with complex coefficients are sometimes expressed in this form, however I'm not sure if/how that would be useful to solve this problem.

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Show that a function of the form  x  (  t  )  =      K    1    cos  ⁡  β  t  +      K    2    sin  ⁡  β  tCan be written as  x  (  t  )  =  K  cos  ⁡  (  B  t  −  ϕ  )Where   K  =            K      1      2        +          K      2      2      I know that linear systems with complex coefficients are sometimes expressed in this form, however I&#039;m not sure if/how that would be useful to solve this problem.

anclarlo5h12v · Accepted Answer

x  (  t  )  =  K      (                  K        1            K        cos    ⁡    (    β    t    )    +                  K        2            K        sin    ⁡    (    β    t    )    )  Since   K  ≥      K    1    ,      K    2   we get             K      1        K    ,            K      2        K    ≤  1, and since             (                        K          1                K            )        2    +            (                        K          2                K            )        2    =  1 I can choose an angle   ϕ such that   cos  ⁡  (  ϕ  )  =            K      1        K   and   sin  ⁡  (  ϕ  )  =            K      2        K  . So then  x  (  t  )  =  K  (  cos  ⁡  (  ϕ  )  cos  ⁡  (  β  t  )  +  sin  ⁡  (  ϕ  )  sin  ⁡  (  β  t  )  )  =  K  cos  ⁡  (  β  t  −  ϕ  )Where the last equality holds by the subtraction formula for cosine.

Show that a function of the form x ( t ) = K 1 </msub> cos

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