Complementary Angles
GeometryComplementary 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 $90^\circ$. You can think of them as two pieces that together complete a right angle corner.
Example
A $30^\circ$ angle and a $60^\circ$ angle are complementary ($30 + 60 = 90$). A $45^\circ$ angle is its own complement. If one angle is $25^\circ$, its complement is $90 - 25 = 65^\circ$.
Key Insight
Complementary comes from "complete" - two complementary angles complete a right angle. A memory trick: C in Complementary goes with C in Corner ($90^\circ$ corner).
Definition
Two angles are complementary if and only if their measures sum to $90^\circ$. Complementary angles do not need to be adjacent (side-by-side); they just need to sum to $90^\circ$. In a right triangle, the two acute angles are always complementary.
Example
Angles of $37^\circ$ and $53^\circ$ are complementary ($37 + 53 = 90$). In a right triangle with one acute angle of $28^\circ$, the other acute angle is $90 - 28 = 62^\circ$. The sine of an angle equals the cosine of its complement: $\sin(30^\circ) = \cos(60^\circ)$.
Key Insight
The co-function identities in trigonometry arise directly from complementary angles: $\sin\theta = \cos(90^\circ - \theta)$, $\tan\theta = \cot(90^\circ - \theta)$. This is why "cosine" literally means "complement's sine" - it is the sine of the complementary angle.
Definition
Angles $\alpha$ and $\beta$ are complementary iff $\alpha + \beta = \pi/2$. The co-function identities $\sin(\pi/2 - x) = \cos x$, $\tan(\pi/2 - x) = \cot x$, $\sec(\pi/2 - x) = \csc x$ are the analytic expression of complementarity. In a right triangle with legs $a, b$ and hypotenuse $c$, the two acute angles are complementary and their trig values are related by these identities.
Example
If $\tan\theta = 3/4$, then $\tan(90^\circ - \theta) = \cot\theta = 4/3$. This swap of opposite and adjacent sides reflects the complementary relationship between the two acute angles in the right triangle.
Key Insight
Complementary angles appear in the symmetry of the unit circle: $\sin(\pi/2 - \theta) = \cos\theta$ reflects the symmetry of the circle about the line $y = x$. This symmetry underpins the interchange between sine and cosine in Fourier analysis and signal processing.