What are Trigonometric Ratios?
Trigonometric Ratios are the mathematical relationships that connect the acute angles of a right-angled triangle to the lengths of its sides. The word trigonometry comes from Greek words meaning "triangle measurement." These ratios allow us to calculate unknown distances or angles simply by using fractions formed by the sides of a triangle.
In a right-angled triangle, we name the three sides based on their position relative to a chosen target angle, which we usually call theta (written as a symbol or spelled out). The three sides are:
- Hypotenuse: The absolute longest side, always located directly across from the 90-degree right angle.
- Perpendicular (or Opposite): The side that stands directly facing opposite our chosen angle theta.
- Base (or Adjacent): The side that lies next to our angle theta, running between theta and the right angle.
Imagine you are looking up at the top of a giant smartphone screen resting on a table. The straight height of the screen from the table is the perpendicular, the flat distance along the table from your eyes to the screen base is the base, and your straight line of sight up to the top corner is the hypotenuse.
There are exactly six distinct trigonometric ratios. Three are primary ratios, and three are reciprocal ratios (meaning they are flipped upside down):
| Ratio Name | Abbreviation | Fractional Definition | Reciprocal Relationship |
|---|---|---|---|
| **Cosine** | cos | Base / Hypotenuse | 1 / sec |
| **Tangent** | tan | Perpendicular / Base | 1 / cot |
| **Cosecant** | csc | Hypotenuse / Perpendicular | 1 / sin |
| **Secant** | sec | Hypotenuse / Base | 1 / cos |
| **Cotangent** | cot | Base / Perpendicular | 1 / tan |