Complementary Angles
Complementary angles are two angles whose measures add up to exactly 90 degrees.
Formula
\text{angle } A + \text{angle } B = 90^\circ
Definition
Two angles are complementary if they add up to $90^\circ$ ($\alpha + \beta = \pi/2$), like two pieces that together complete a right-angle corner; they do not need to be adjacent, they just need to sum to $90^\circ$, and in a right triangle the two acute angles are always complementary. This is the geometric root of the co-function identities: $\sin\theta = \cos(90^\circ - \theta)$, $\tan\theta = \cot(90^\circ - \theta)$, $\sec(\pi/2-x)=\csc x$, since "cosine" literally means "complement's sine."
Example
A $30^\circ$ angle and a $60^\circ$ angle are complementary ($30+60=90$), and a $45^\circ$ angle is its own complement; if one angle is $25^\circ$, its complement is $65^\circ$. In a right triangle with one acute angle of $28^\circ$, the other is $62^\circ$, and $\sin(30^\circ) = \cos(60^\circ)$. If $\tan\theta = 3/4$, then $\tan(90^\circ-\theta) = \cot\theta = 4/3$, the swap of opposite and adjacent sides reflecting the complementary relationship.
Key Insight
A memory trick: C in Complementary goes with C in Corner ($90^\circ$). Complementary angles also appear in the symmetry of the unit circle: $\sin(\pi/2 - \theta) = \cos\theta$ reflects the circle's symmetry about the line $y=x$, an interchange between sine and cosine that underpins Fourier analysis and signal processing.