# Trigonometric Functions — Formulas, Graphs, Values

[Trigonometric](/content/tag/trigonometric/index.html)

TL;DR

The six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — turn angles into ratios. Sine and cosine are read from the yyy- and xxx-coordinates of a point on the unit circle; tangent is their ratio; the other three are reciprocals. All six are periodic, defined for most real angles, and form the foundation of trig identities, calculus, and wave physics.

## A Lookup Table That Outgrew Itself

Around 130 BCE, Hipparchus of Rhodes built the first table of chords — astronomical data for converting angles into measured ratios. His tables were physical artefacts: scrolls listing values to two decimal places. By the year 2000, the same lookup happened ten billion times a second inside every GPS satellite. The six trigonometric functions are what Hipparchus's table became when calculus and computers got hold of it.

## What Are Trigonometric Functions?

**Trigonometric functions** are six functions that take an angle as input and return a ratio. They are _defined_ in two equivalent ways:

- **Right-triangle definition** (for acute angles): ratios of side lengths in a right triangle.
- **Unit-circle definition** (for any real angle): coordinates of a point reached by rotating from the positive xxx-axis.

Three are "primary":

- \( 	ext{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)  
- \( 	ext{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)  
- \( 	ext{tan} \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Three are "reciprocal":

- \( 	ext{csc} \theta = \frac{1}{\sin \theta} \)  
- \( 	ext{sec} \theta = \frac{1}{\cos \theta} \)  
- \( 	ext{cot} \theta = \frac{1}{\tan \theta} \)

The mnemonic **SOH-CAH-TOA** (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent) handles the first three. The other three are just "flip it."

## The Six Trigonometric Function Formulas

| Function | Right-triangle | Unit-circle |
| --- | --- | --- |
| sinθ | \( \frac{\text{opp}}{\text{hyp}} \) | yyy-coordinate of (cosθ,sinθ) |
| cosθ | \( \frac{\text{adj}}{\text{hyp}} \) | xxx-coordinate of (cosθ,sinθ) |
| tanθ | \( \frac{\text{opp}}{\text{adj}} \) | \( \frac{\sin \theta}{\cos \theta} \) |
| cscθ | \( \frac{\text{hyp}}{\text{opp}} \) | \( \frac{1}{\sin \theta} \) |
| secθ | \( \frac{\text{hyp}}{\text{adj}} \) | \( \frac{1}{\cos \theta} \) |
| cotθ | \( \frac{\text{adj}}{\text{opp}} \) | \( \frac{\cos \theta}{\sin \theta} \) |

## Trigonometric Function Values at Standard Angles

The five "exact" angles you should know cold:

| θ | 0 | \( \frac{\pi}{6} \) (30°) | \( \frac{\pi}{4} \) (45°) | \( \frac{\pi}{3} \) (60°) | \( \frac{\pi}{2} \) (90°) |
| --- | --- | --- | --- | --- | --- |
| sinθ | 0 | \( \frac{1}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{3}}{2} \) | 1 |
| cosθ | 1 | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{1}{2} \) | 0 |
| tanθ | 0 | \( \frac{1}{\sqrt{3}} \) | 1 | \( \sqrt{3} \) | undefined |
| cscθ | undefined | 2 | \( \sqrt{2} \) | \( \frac{2}{\sqrt{3}} \) | 1 |
| secθ | 1 | \( \frac{2}{\sqrt{3}} \) | \( \sqrt{2} \) | 2 | undefined |
| cotθ | undefined | \( \sqrt{3} \) | 1 | \( \frac{1}{\sqrt{3}} \) | 0 |

## Trig Functions in Four Quadrants

On the unit circle, signs depend on quadrant. The mnemonic **ASTC** (" **A** ll **S** tudents **T** ake **C** alculus") gives which functions are positive in each quadrant, counter-clockwise from quadrant I:

| Quadrant | Range of θ | Positive functions |
| --- | --- | --- |
| I | 0 to \( \frac{\pi}{2} \) | **A** ll — sin, cos, tan all positive |
| II | \( \frac{\pi}{2} \) to π | **S** ine only |
| III | π to \( \frac{3\pi}{2} \) | **T** angent only (and cot) |
| IV | \( \frac{3\pi}{2} \) to 2π | **C** osine only (and sec) |

## The Graphs of the Six Trigonometric Functions

### Sine and cosine

Both are smooth waves with amplitude 1 and period \( 2\pi \). They look identical, just shifted: \( 	ext{cos}\,x = 	ext{sin}(x + \frac{\pi}{2}) \) — cosine leads sine by a quarter period.

- Domain: all real numbers
- Range: \[-1, 1\]
- Period: \( 2\pi \)

### Tangent and cotangent

Both have asymptotes — vertical lines the graph never touches.

- \( 	ext{tan}\,x \) asymptotes at \( x = \frac{\pi}{2} + n\pi \); period \( \pi \); range all reals.

### Secant and cosecant

Both look like upside-down U-shapes between asymptotes.

- \( 	ext{sec}\,x \) asymptotes where \( 	ext{cos} = 0 \); range \((-\infty, -1] \cup [1, \infty)\).
- \( 	ext{csc}\,x \) asymptotes where \( 	ext{sin} = 0 \); range \((-\infty, -1] \cup [1, \infty)\).

## Domain and Range of Trigonometric Functions

| Function | Domain | Range |
| --- | --- | --- |
| sinθ | (−∞, ∞) | \[-1, 1\] |
| cosθ | (−∞, ∞) | \[-1, 1\] |
| tanθ | \( θ ≠ \frac{\pi}{2} + n\pi \) | (−∞, ∞) |
| cscθ | \( θ ≠ n\pi \) | \((-\infty,−1] \cup [1,∞)\) |
| secθ | \( θ ≠ \frac{\pi}{2} + n\pi \) | \((-\infty,−1] \cup [1,∞)\) |
| cotθ | \( θ ≠ n\pi \) | (−∞, ∞) |

## Trigonometric Function Identities

The major identity families that connect the six functions:

- **Pythagorean.** \( 	ext{sin}^2 + 	ext{cos}^2 = 1 \);
- **Reciprocal.**  \( \text{csc} = \frac{1}{\text{sin}} \);
- **Quotient.** \( \text{tan} = \frac{\text{sin}}{\text{cos}} \);
- **Even/odd.** \( 	ext{sin}(-x) = -\text{sin} \);

## Key Takeaways

- The six **trigonometric functions** are sine, cosine, tangent, cosecant, secant, and cotangent — defined on a unit circle, generalising the right-triangle ratios.
- Sine and cosine have domain all reals; the other four have domain gaps where division by zero would happen.
- The exact values at 0, π/6, π/4, π/3, and π/2 follow the pattern for sine and cosine.
- ASTC tells you the sign of each function in each quadrant — without it, sign errors dominate.