Sine of Complement equals Cosine

From ProofWiki
Jump to navigation Jump to search

Theorem

$\sin \left({\dfrac \pi 2 - \theta}\right) = \cos \theta$

where $\sin$ and $\cos$ are sine and cosine respectively.

That is, the cosine of an angle is the sine of its complement.


Proof 1

\(\displaystyle \sin \left({\frac \pi 2 - \theta}\right)\) \(=\) \(\displaystyle \sin \frac \pi 2 \cos \theta - \cos \frac \pi 2 \sin \theta\) Sine of Difference
\(\displaystyle \) \(=\) \(\displaystyle 1 \times \cos \theta - 0 \times \sin \theta\) Sine of Right Angle and Cosine of Right Angle
\(\displaystyle \) \(=\) \(\displaystyle \cos \theta\)

$\blacksquare$


Proof 2

\(\displaystyle \sin \left({\frac \pi 2 - \theta}\right)\) \(=\) \(\displaystyle - \sin \left({\theta - \frac \pi 2}\right)\) Sine Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle \cos \left({\theta - \frac \pi 2 + \frac \pi 2}\right)\) Cosine of Angle plus Right Angle
\(\displaystyle \) \(=\) \(\displaystyle \cos \theta\)

$\blacksquare$


Proof 3

\(\displaystyle \map \sin {\dfrac \pi 2 - \theta}\) \(=\) \(\displaystyle \map \Im {e^{i \paren {\frac \pi 2 - \theta} } }\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle \map \Im {e^{i \frac \pi 2} e^{-i \theta} }\) Exponent Combination Laws
\(\displaystyle \) \(=\) \(\displaystyle \map \Im {\paren {\cos \dfrac \pi 2+i \sin \dfrac \pi 2} e^{-i \theta} }\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle \map \Im {i e^{-i \theta} }\) Cosine of Right Angle, Sine of Right Angle
\(\displaystyle \) \(=\) \(\displaystyle \map \Re {e^{-i \theta} }\)
\(\displaystyle \) \(=\) \(\displaystyle \map \cos {-\theta}\) Euler's Formula
\(\displaystyle \) \(=\) \(\displaystyle \cos \theta\) Cosine Function is Even

$\blacksquare$


Also see


Sources