The equation is a cos ⁡ ( ω t ) + b sin ⁡ ( ω t ) = A cos ⁡ ( ω t + ϕ ) I must have forgotten how the right hand side follows from the left hand side.Some hints would be helpful

Question

The equation is  a  cos  ⁡      (    ω    t    )    +  b  sin  ⁡      (    ω    t    )    =  A  cos  ⁡      (    ω    t    +    ϕ    )  I must have forgotten how the right hand side follows from the left hand side.Some hints would be helpful

Taniya Wood · Accepted Answer

Let   A  =            a      2        +          b      2       and   ϕ be an angle such that   cos  ⁡  ϕ  =            a                        a          2                +                  b          2                     and   sin  ⁡  ϕ  =                    −        b                              a          2                +                  b          2                    Then, by using the identity   cos  ⁡  (  x  +  y  )  =  cos  ⁡  x  cos  ⁡  y  −  sin  ⁡  x  sin  ⁡  y, we have:                    a        cos        ⁡        (        ω        t        )        +        b        sin        ⁡        (        ω        t        )                            =                              a            2                    +                      b            2                                    [                                    a                                                a                  2                                +                                  b                  2                                                              cos          ⁡          (          ω          t          )          −                                                    −                b                                                              a                  2                                +                                  b                  2                                                              sin          ⁡          (          ω          t          )          ]                                                  =        A                  [          cos          ⁡          ϕ          cos          ⁡          (          ω          t          )          −          sin          ⁡          ϕ          sin          ⁡          (          ω          t          )          ]                                                  =        A        cos        ⁡        (        ω        t        +        ϕ        )        .

Hailee Stout · Accepted Answer

Try breaking down   A  cos  ⁡  (  ω  t  +  ϕ  )                    A        cos        ⁡        (        ω        t        +        ϕ        )                    =                    A        (        cos        ⁡        ω        t        cos        ⁡        ϕ        −        sin        ⁡        ω        t        sin        ⁡        ϕ        )                                  =                    (        A        cos        ⁡        ϕ        )        cos        ⁡        ω        t        −        (        A        sin        ⁡        ϕ        )        sin        ⁡        ω        t            Try to show that for arbitrary constants a and b that there exists constants A and   ϕ such that  a  =  A  cos  ⁡  ϕand  b  =  A  sin  ⁡  ϕWhat is       a    2    +      b    2  ?

The equation is a cos ⁡ ( ω t

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