Trigonometry
Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles.
Definition
Trigonometry is the branch of mathematics that studies triangles, especially the relationships between their angles and sides, letting you find a missing side or angle from known measurements. Formally, it defines and studies the six trigonometric functions, sine, cosine, tangent, cosecant, secant, and cotangent, which relate the angles of a right triangle to ratios of its sides. These ratios extend beyond right triangles to the unit circle, where for any angle $t$, $\cos(t)$ and $\sin(t)$ are defined as the $x$- and $y$-coordinates of the point $(\cos t, \sin t)$ on the circle. The resulting functions are periodic and smooth, and satisfy identities such as $\sin^2(t) + \cos^2(t) = 1$.
Example
If a ramp makes a $30$-degree angle with the ground and you know its length, trigonometry lets you find the height of its top without measuring it directly. More precisely, for a right triangle with an angle of $40^\circ$ and a hypotenuse of $10$ units, $\sin(40^\circ) = \text{opposite}/10$, so the opposite side $= 10 \times \sin(40^\circ) \approx 6.43$ units. This same idea reaches into complex analysis through Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$, which shows that sine and cosine are the imaginary and real parts of the complex exponential.
Key Insight
The word "trigonometry" comes from Greek words meaning "triangle measurement," and it began with ancient astronomers who needed to calculate distances to stars. It bridges geometry and algebra and has grown into a foundation for calculus, physics, and engineering. Via Fourier analysis, every periodic function can be decomposed into sums of sines and cosines, making trigonometry the underlying language of signal processing, quantum mechanics, and differential equations.