Complete guide with memory aids, graphs, and interactive quiz
Trigonometric functions relate angles of a triangle to the lengths of its sides. The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
Use the mnemonic “SOH-CAH-TOA” for right triangles:
The reciprocal functions are: Cosecant = 1/Sine, Secant = 1/Cosine, Cotangent = 1/Tangent.
Understanding the domain, range, and period of each function is essential for solving trigonometric equations and graphing.
The set of all real numbers is denoted by ℝ = (−∞, +∞) = ]−∞, ∞[.
All trigonometric functions except tangent and cotangent have domains of all real numbers (with some exceptions for secant and cosecant).
Here’s a complete reference table for all six trigonometric functions:
| Function | Domain | Range | Period |
|---|---|---|---|
| Sine (sin x) | ℝ (all real numbers) | [-1, 1] | 2π |
| Cosine (cos x) | ℝ (all real numbers) | [-1, 1] | 2π |
| Tangent (tan x) | x ≠ π/2 + nπ, n∈ℤ | ℝ (all real numbers) | π |
| Cotangent (cot x) | x ≠ nπ, n∈ℤ | ℝ (all real numbers) | π |
| Secant (sec x) | x ≠ π/2 + nπ, n∈ℤ | (-∞, -1] ∪ [1, ∞) | 2π |
| Cosecant (csc x) | x ≠ nπ, n∈ℤ | (-∞, -1] ∪ [1, ∞) | 2π |
TANGENT is undefined at 90° (π/2) and every 180° (π) from there: π/2, 3π/2, 5π/2…
COTANGENT is undefined at 0°, 180° (π), 360° (2π)… i.e., at nπ
SECANT has same exclusions as tangent (it’s the reciprocal of cosine)
COSECANT has same exclusions as cotangent (it’s the reciprocal of sine)
Note i: If a trigonometric function is multiplied by a constant ‘K’ (e.g., K·sin x), the range changes but the domain remains the same.
Example: For f(x) = 3 sin x, domain = ℝ, range = [-3, 3]
Note ii: If the angle of a trigonometric function is multiplied by a constant ‘K’ (e.g., sin(Kx)), the domain changes but the range remains the same.
Example: For f(x) = sin(2x), domain = ℝ, range = [-1, 1] (same as sin x)
1. f(x) = 3 sin x: Domain = ℝ, Range = [-3, 3]
2. f(x) = tan(x/2): Domain = x ≠ π + 2nπ, Range = ℝ
3. f(x) = cos(3x): Domain = ℝ, Range = [-1, 1]
A function f(x) is periodic if there exists a positive number P such that f(x + P) = f(x) for all x in the domain.
The smallest such P is called the fundamental period or simply the period.
All trigonometric functions are periodic. Their graphs repeat at regular intervals.
Sine and Cosine: Period = 2π (360°)
Tangent and Cotangent: Period = π (180°)
Secant and Cosecant: Period = 2π (360°)
1. Period of sin(ax + b) or cos(ax + b) = 2π/|a|
2. Period of tan(ax + b) or cot(ax + b) = π/|a|
3. Period of sec(ax + b) or csc(ax + b) = 2π/|a|
For f(ax + b):
Sine/Cosine family: Period = (Standard Period) / |a|
Tangent/Cotangent family: Period = (Standard Period) / |a|
Where “Standard Period” is 2π for sin/cos/sec/csc and π for tan/cot.
Graphs help visualize the behavior of trigonometric functions:
Sine Graph: Starts at origin (0,0), goes up to 1 at π/2, back to 0 at π, down to -1 at 3π/2, back to 0 at 2π. Smooth wave.
Cosine Graph: Starts at (0,1), goes to 0 at π/2, to -1 at π, back to 0 at 3π/2, to 1 at 2π. Also a smooth wave but shifted.
Tangent Graph: Has vertical asymptotes at π/2, 3π/2, etc. Passes through origin. Repeats every π.
From graphs, you can easily determine domain (x-values with no breaks), range (y-values covered), and period (horizontal length of one complete cycle).
Test your knowledge with these 50 multiple choice questions. Select your answer and check the key at the end.
Connect trigonometric functions to real-world applications: sine waves in sound and light, tangent in slopes and angles, periodic functions in seasons and tides. This makes the concepts more memorable and meaningful.