# Understanding Sin, Cos, and Tan in Trigonometry

This page provides a comprehensive explanation of the fundamental trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios are essential for solving problems involving right-angled triangles in trigonometry.

**Definition**: **Sin** is the ratio of the leg opposite of the given angle to the hypotenuse.

**Definition**: **Cos** is the ratio of the adjacent leg to the hypotenuse (relative to the given angle).

**Definition**: **Tan** is the ratio of the leg opposite of the given angle to the adjacent leg (relative to the given angle).

The page includes a diagram of a right-angled triangle, labeling the sides as adjacent, opposite, and hypotenuse in relation to a given angle 'a'. This visual aid helps in understanding how these ratios are derived from the triangle's components.

**Highlight**: A crucial point to remember is that the right angle should not be used as the given angle when applying these ratios.

**Vocabulary**: The **hypotenuse** is the longest side of a right-angled triangle, opposite the right angle.

To help memorize these ratios, the page introduces the acronym "SOH CAH TOA":

- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent

**Example**: A practical application of the tangent ratio is demonstrated with a step-by-step problem-solving approach:

- Identify the sides of the triangle.
- Choose the appropriate ratio (in this case, tan, as we're solving for y and already have the opposite side).
- Set up the problem using tan of the given angle, the known side, and the variable.
- Solve the equation.

The example shows how to find an unknown side length (y) using the tangent of a 22° angle in a right-angled triangle with a known adjacent side of 40 units.

**Quote**: "Some old hippie, caught another hippie tripping on acid." This mnemonic sentence is provided as an alternative way to remember SOH CAH TOA.

By understanding and applying these **sin, cos, tan formulas**, students can effectively solve a wide range of trigonometric problems involving right-angled triangles. This foundational knowledge is crucial for more advanced topics in trigonometry and its applications in various fields of science and engineering.