In a book I'm reading, a claim is made that f ( a ) =...

enrotlavaec

enrotlavaec

Answered

2022-06-26

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...

Answer & Explanation

Anika Stevenson

Anika Stevenson

Expert

2022-06-27Added 19 answers

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

Semaj Christian

Expert

2022-06-28Added 12 answers

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

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