# Trigonometry — Complete Guide to Formulas & Identities

## TL;DR  
This trigonometry complete guide covers the six trig functions (sin, cos, tan, csc, sec, cot), the unit circle, the four families of identities (Pythagorean, reciprocal, sum-and-difference, double-angle), the six inverse trig functions and the standard derivatives.

## A Discipline Older Than Greek Geometry  
Babylonian astronomers tracking the moon's position around 1800 BCE were already using chord-length tables — the direct ancestor of sine.  
By Hipparchus in 130 BCE, the tables were systematic. By Aryabhata in 499 CE, sine and cosine had distinct names. By Euler in 1748, the six trig functions were modern analytic objects with derivatives, series expansions, and a connection to complex numbers via eiθ=cos⁡θ+isin⁡θ. Trigonometry is the oldest continuously-developed branch of mathematics — 3,800 years and counting.

## What Is Trigonometry?  
**Trigonometry** is the branch of mathematics that studies relationships between angles and side-length ratios in triangles, extended through the unit circle to all real angles. Six functions — sine, cosine, tangent, cosecant, secant, cotangent — convert angles into ratios. A handful of identities convert between expressions involving those functions. From this scaffold, the entire mathematics of waves, oscillations, rotations, and complex analysis follows.

Three perspectives, one subject:  
- **Right-triangle perspective.** Trig ratios are ratios of side lengths in a right triangle.  
- **Unit-circle perspective.** Trig functions are coordinates of points reached by rotating along the unit circle.  
- **Analytic perspective.** Trig functions are smooth, periodic functions with derivatives, series expansions, and a deep connection to e^x.

This guide moves through all three.

## The Six Trigonometric Functions  
| Function        | Right-triangle                     | Unit-circle                       |
|-----------------|-----------------------------------|-----------------------------------|
| sin⁡θ          | \frac{\text{opp}}{\text{hyp}}  | y-coordinate of (cos⁡θ,sin⁡θ)   |
| cos⁡θ          | \frac{\text{adj}}{\text{hyp}}  | x-coordinate of (cos⁡θ,sin⁡θ)   |
| tan⁡θ          | \frac{\text{opp}}{\text{adj}}  | \frac{\sin\theta}{\cos\theta}|
| csc⁡θ          | \frac{\text{hyp}}{\text{opp}}  | \frac{1}{\sin\theta}          |
| sec⁡θ          | \frac{\text{hyp}}{\text{adj}}  | \frac{1}{\cos\theta}          |
| cot⁡θ          | \frac{\text{adj}}{\text{opp}}  | \frac{\cos\theta}{\sin\theta}|

**SOH-CAH-TOA** gives the first three from the right triangle. The other three are reciprocals: csc⁡=1/sin⁡, sec⁡=1/cos⁡, cot⁡=1/tan⁡.

## The Unit Circle and Standard Angles  
The unit circle is centred at the origin with radius 1. A point at angle θ from the positive x-axis has coordinates (cos⁡θ,sin⁡θ).  
The five "exact" angles every student must know cold:

| θ           | 0           | \frac{\pi}{6}       | \frac{\pi}{4}       | \frac{\pi}{3}       | \frac{\pi}{2}       |
|-------------|-------------|---------------------|---------------------|---------------------|---------------------|
| sin⁡θ      | 0           | \frac{1}{2}        | \frac{\sqrt{2}}{2} | \frac{\sqrt{3}}{2} | 1                   |
| cos⁡θ      | 1           | \frac{\sqrt{3}}{2}| \frac{\sqrt{2}}{2} | \frac{1}{2}        | 0                   |
| tan⁡θ      | 0           | \frac{1}{\sqrt{3}}| 1                   | \sqrt{3}           | undefined           |

**The memorisation trick.** Sine's row is 0/2,1/2,2/2,3/2,4/2 — indexed by square roots of 0 through 4. Cosine runs the same sequence backwards.
For angles in quadrants II, III, IV, use **reference angles + ASTC**.

## The Four Identity Families  
Trigonometric identities collapse complex expressions into simpler ones. The four major families:

### Pythagorean identities  
sin⁡2θ+cos⁡2θ=1  
1+tan⁡2θ=sec⁡2θ  
1+cot⁡2θ=csc⁡2θ

All three follow from the unit-circle equation x²+y²=1.

### Sum-and-difference identities  
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B  
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B  
tan⁡(A±B)=\frac{tan⁡A±tan⁡B}{1∓tan⁡Atan⁡B}

These identities are the most-used in trigonometric problem-solving — they derive from them.

### Double-angle identities  
sin⁡(2θ)=2sin⁡θcos⁡θ  
cos⁡(2θ)=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1  
tan⁡(2θ)=\frac{2tan⁡θ}{1−tan²θ}

### Sum-to-product (and product-to-sum) identities  
sin⁡A+sin⁡B=2sin⁡(\frac{A+B}{2})cos⁡(\frac{A−B}{2})  
cos⁡A−cos⁡B=−2sin⁡(\frac{A+B}{2})sin⁡(\frac{A−B}{2})

## The Six Inverse Trigonometric Functions  
| Inverse function | Domain            | Range (principal)        |
|-------------------|-------------------|--------------------------|
| arcsin⁡x          | [-1,1]            | [-π/2,π/2]              |
| arccos⁡x          | [-1,1]            | [0,π]                   |
| arctan⁡x          | (-∞,∞)           | (-π/2,π/2)              |
| arccsc⁡x          | |x|≥1              | [-π/2,0)∪(0,π/2)      |
| arcsec⁡x          | |x|≥1              | [0,π/2)∪(π/2,π)      |
| arccot⁡x          | (-∞,∞)           | (0,π)                   |

## Trigonometric Derivatives  
| Function         | Derivative          |
|------------------|---------------------|
| sin⁡x           | cos⁡x               |
| cos⁡x           | −sin⁡x              |
| tan⁡x           | sec²x               |
| cot⁡x           | −csc²x              |
| sec⁡x           | sec⁡xtan⁡x         |
| csc⁡x           | −csc⁡xcot⁡x         |

**The "co-" rule.** Every "co-" function has a negative sign in its derivative. The three "non-co" functions don't.

## Three Worked Examples — Quick, Standard, Stretch  
### Quick  
**Evaluate sin⁡(π/3)+cos⁡(π/6).**  
From the standard-angle table: sin⁡(π/3)=√3/2 and cos⁡(π/6)=√3/2.  
sin⁡(π/3)+cos⁡(π/6)=√3/2+√3/2=√3.

### The Mistake Worth Making Once — Standard Example  
**Prove that \frac{1 − cos⁡(2θ)}{sin⁡(2θ)} = tan⁡θ.**
The wrong path. A student writes... (show steps here)  
So where's the wrong path?

### Stretch  
**Find the exact value of sin⁡(15°).**  
15°=45°−30°, so... (show steps here)

## Why Trigonometry Matters Beyond the Triangle  
Trigonometry is the language of every periodic phenomenon in the physical world.

- **GPS.**  
- **AC electricity.**  
- **Music.**  
- **Computer graphics.**

## The Mathematicians Who Built Trigonometry  
Trigonometry is the work of more than a hundred named contributors across 3,800 years. Four matter most.
- **Hipparchus of Nicaea**...  
- **Aryabhata**...  
- **al-Khwarizmi**...  
- **Leonhard Euler**...

**Why it matters:**

## Where Solutions Go Off the Rails — Common Mistakes  
### **Mistake 1: Working in degrees when the formula assumes radians**  
### **Mistake 2: Forgetting to use the reference angle**  
### **Mistake 3: Confusing sin⁡−1x with 1/sin⁡x**

## Key Takeaways  
- **Trigonometry** is the study of relationships between angles and side ratios.  
- The six trig functions are sine, cosine, tangent and their three reciprocals.  
- The four identity families — Pythagorean, sum-and-difference, double-angle, and sum-to-product — convert between forms.  
- Inverse trig functions, restricted to principal branches, recover angles from ratios.  
- Trigonometry powers GPS, AC electricity, music, computer graphics, astronomy, and engineering.