Sine of Complement equals Cosine

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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

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

$\blacksquare$


Also see


Sources