Trigonometric Functions — Formulas, Graphs, Values
Trigonometric Functions — Formulas, Graphs, Values
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.