Sine and Cosine are Cofunctions

Theorem

The sine and cosine are cofunctions:

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

\(\ds \sin x\)

\(=\)

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

\(\ds \cos x\)

\(=\)

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

Proof

We have:

Sine of Complement equals Cosine

Cosine of Complement equals Sine

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