In a book I'm reading, a claim is made that f ( a ) = r cos ⁡ ( a ) − m sin ⁡ ( a )has the maximum r 2 + m 2 (where a, r and m are real numbers).But I'm not sure how to prove it or even if it's true. Setting d d a f ( a ) = 0 gives − r sin ⁡ ( a ) − m cos ⁡ ( a ) = 0 or tan ⁡ ( a ) = − m / r, so a = arctan ⁡ ( − m / r ). Plugging that in into wolfram does not give the claimed result...

Question

In a book I&#039;m reading, a claim is made that  f  (  a  )  =  r  cos  ⁡  (  a  )  −  m  sin  ⁡  (  a  )has the maximum            r      2        +          m      2      (where a, r and m are real numbers).But I&#039;m not sure how to prove it or even if it&#039;s true. Setting             d              d        a              f  (  a  )  =  0 gives   −  r  sin  ⁡  (  a  )  −  m  cos  ⁡  (  a  )  =  0 or   tan  ⁡  (  a  )  =  −  m      /    r, so   a  =  arctan  ⁡  (  −  m      /    r  ). Plugging that in into wolfram does not give the claimed result...

Anika Stevenson · Accepted Answer

f  (  a  )  =  r  cos  ⁡  (  a  )  −  m  sin  ⁡  (  a  )  =            r      2        +          m      2        (      r                  r        2            +              m        2              cos  ⁡  a  −      m                  r        2            +              m        2              cos  ⁡  a  )Now suppose   cos  ⁡  θ  =      r                  r        2            +              m        2            ,   sin  ⁡  θ  =      m                  r        2            +              m        2            , then you have   f  (  a  )  =            r      2        +          m      2        cos  ⁡  (  a  +  θ  ), from which the maximum and minimum values are obvious.

Semaj Christian · Accepted Answer

From your link,            m      2              r                                    m            2                                r            2                          +        1              +      r                            m          2                          r          2                    +      1        =            m      2                      m        2            +              r        2              +            r      2                      m        2            +              r        2              =                    m        2            +              r        2                            m        2            +              r        2              =            m      2        +          r      2