----- > [!proposition] Proposition. ([[pythagorean trig identity]]) > For $\omega \in \rr$ we have $\cos ^{2} \omega + \sin ^{2} \omega = 1$. > [!proof]- Proof. ([[pythagorean trig identity]]) > Consider a [[vector]] $z$ on unit circle in the complex plane sweeping out an angle of $\omega$. ![[CleanShot 2023-01-10 at 23.54.11.jpg|200]] Clearly $z=\cos \omega + j\sin \omega$. Now take the [[modulus]] and the result is immediate: $1=|z |^{2}= \cos ^{2}\omega + \sin ^{2} \omega.$ Alternatively we can consider a complex exponential with [[modulus]] $1$ and angle $\omega$, $e^{j \omega}$ , and convert to polar form via [[Euler's formula]] to obtain $e^{j \omega}= \cos \omega + j\sin \omega$ and the result follows by taking the squared [[modulus]] of both sides. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```