Trigonometry explores the relationships between the angles and lengths of triangles. Explore the core concepts below through clean explanations, visual diagrams, tables, and graphs.
Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics focused on the relationships between side lengths and angles of triangles. It forms the foundation for advanced mathematics, physics, astronomy, geospatial mapping, and computer graphics.
A right-angled triangle consists of a 90° angle, a selected reference angle (represented by the Greek letter theta, θ), and three standard sides relative to that angle.
The labels assigned to the triangle's sides depend entirely on their orientation relative to the designated reference angle θ.
| Side | Description |
|---|---|
| Hypotenuse (H) | The longest side. Always directly opposite the 90° right angle. |
| Opposite (O) | The side directly across from the target reference angle θ. |
| Adjacent (A) | The side situated next to the target reference angle θ that is not the hypotenuse. |
The three main functions relate the reference angle to specific fractional ratios of the triangle's sides.
| Function | Mathematical Notation | Mnemonic |
|---|---|---|
| Sine (sin θ) |
Opposite
Hypotenuse
|
SOH |
| Cosine (cos θ) |
Adjacent
Hypotenuse
|
CAH |
| Tangent (tan θ) |
Opposite
Adjacent
|
TOA |
These functions represent the mathematical inverses of the primary ratios, yielding reciprocal fractions of the sides.
| Function | Mathematical Notation | Equivalent To |
|---|---|---|
| Cosecant (csc θ) |
Hypotenuse
Opposite
|
1
sin θ
|
| Secant (sec θ) |
Hypotenuse
Adjacent
|
1
cos θ
|
| Cotangent (cot θ) |
Adjacent
Opposite
|
1
tan θ
|
Unlike discrete angles in a right-angled triangle, trigonometric functions map continuously across an infinite domain. Below is the smooth, cyclical wave of y = sin(θ) over a standard 0° to 360° range.
Specific key angles yield exact, geometric outputs. These values are essential to recognize and memorize for mathematical computations.
| Angle (θ) | 0° (0) | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) |
|---|---|---|---|---|---|
| Sin θ | 0 |
12
|
√22
|
√32
|
1 |
| Cos θ | 1 |
√32
|
√22
|
12
|
0 |
| Tan θ | 0 |
√33
|
1 | √3 | Undefined |
Depending on which quadrant an angle falls in on the Cartesian plane, the outputs of the trigonometric ratios can transition between positive and negative configurations.
| Quadrant | Angle Range | Positive Functions |
|---|---|---|
| Quadrant I | 0° to 90° | All (sin, cos, tan) |
| Quadrant II | 90° to 180° | Sin (and csc) |
| Quadrant III | 180° to 270° | Tan (and cot) |
| Quadrant IV | 270° to 360° | Cos (and sec) |