Trigonometric Functions

Trigonometry started thousands of years ago as the study of triangles. But the three most famous trig functions — **sine**, **cosine**, and **tangent** — turn out to describe almost everything that repeats in nature. Sound waves, light, electrical signals, planetary orbits, the tides, a child on a swing — all are described by sine and cosine functions. Understanding them is a gateway to physics, engineering, music, and modern technology.

For any right triangle, with an angle θ (theta):\n\n- **sin(θ) = opposite / hypotenuse**\n- **cos(θ) = adjacent / hypotenuse**\n- **tan(θ) = opposite / adjacent**\n\nMnemonic: **SOH-CAH-TOA**.\n\n- Sine = Opposite over Hypotenuse\n- Cosine = Adjacent over Hypotenuse\n- Tangent = Opposite over Adjacent\n\nThese ratios only depend on the angle, not the triangle's size. Two right triangles with the same angle have the same sine, cosine, and tangent — the sides scale together.

To define trig for ANY angle — not just acute ones in a triangle — we use the **unit circle**: a circle of radius 1 centered at the origin.\n\nFor any angle θ measured from the positive x-axis:\n- The point on the unit circle at angle θ has coordinates (cos θ, sin θ).\n- So cos is the x-coordinate, sin is the y-coordinate.\n\nThis extends sine and cosine to all real angles:\n- sin(0°) = 0, sin(90°) = 1, sin(180°) = 0, sin(270°) = −1\n- cos(0°) = 1, cos(90°) = 0, cos(180°) = −1, cos(270°) = 0\n\nBoth functions repeat every 360° (or 2π radians). That's why sine and cosine look like waves when you graph them.

In higher math, we usually measure angles in **radians**, not degrees.\n\n- One full circle = **2π radians** = 360°\n- π radians = 180°\n- π/2 radians = 90°\n- π/3 radians = 60°\n- π/6 radians = 30°\n\nWhy radians? They come from arc length. An angle of θ radians in a circle of radius r subtends an arc of length s = rθ. No conversion factors, no 360. Radians make trig formulas cleaner, especially in calculus.

The graphs of y = sin(x) and y = cos(x) are **waves**:\n\n- Amplitude (height) = 1\n- Period (length of one cycle) = 2π\n- sin starts at (0, 0) and rises; cos starts at (0, 1) and falls\n- They are the SAME wave shifted by π/2 — cos(x) = sin(x + π/2)\n\nFor the more general form **y = A sin(Bx + C) + D**:\n- **A** = amplitude (height of the wave from center)\n- **B** = affects period (period = 2π/B — bigger B means faster oscillation)\n- **C** = horizontal shift\n- **D** = vertical shift (moves the whole wave up or down)\n\nEvery wave you see in nature — sound, light, electrical signals — can be expressed with these four parameters.

Sine and cosine describe anything that repeats:\n\n- **Tides**: h(t) = 5 sin(π t / 12) + 10 models a tide cycling every 24 hours (period = 2π/(π/12) = 24).\n- **Sound**: a pure tone is a sine wave; its frequency is how many cycles per second (Hz).\n- **Seasonal temperature**: average temperatures in your city cycle seasonally and can be modeled with a sine function.\n- **Ferris wheels**: your height as you ride follows sine.\n- **Pendulums**: the angle of a pendulum follows approximately sine over time.\n- **Alternating current (AC)**: the voltage in your wall outlet is a sine wave, 60 Hz in the US.\n\nIf you can identify A, B, C, and D for a real-world phenomenon, you've modeled it.

Trigonometric functions are one of the deepest ideas in mathematics — geometry, algebra, and analysis unified in a single elegant framework. Everything cyclical in nature, from your heartbeat to radio signals, speaks the language of sine and cosine. Once you master them here, you'll meet them again in calculus, complex numbers, differential equations, Fourier analysis, and the inner workings of every modern technology.