Cofunction of Cofunction

Theorem

Let $g$ be a cofunction of $f$.

Then $f$ is a cofunction of $g$.

That is, cofunctions exist in pairs.

Proof

That is:

\(\ds \forall x \in \R: \, \)

\(\ds \map g x\)

\(=\)

\(\ds \map f {90 \degrees - x}\)

Definition of Cofunction

\(\ds \map g {90 \degrees - x}\)

\(=\)

\(\ds \map f {90 \degrees - \paren {90 \degrees - x} }\)

\(\ds \)

\(=\)

\(\ds \map f x\)

Hence the result by definition of cofunction.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cofunctions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cofunctions