Mathematics - Trigonometry - Basics

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles, especially right-angled triangles. The fundamental purpose of trigonometry is to solve problems involving these geometric figures by utilizing specific ratios and functions.

Basic Concepts of Trigonometry

1. Right-Angled Triangle:
   A right-angled triangle is a triangle in which one of the angles is exactly 90 degrees. The side opposite this right angle is called the hypotenuse, and the other two sides are referred to as the adjacent and opposite sides, relative to a given angle.

2. Trigonometric Ratios:
   The primary ratios used in basic trigonometry are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These are defined as follows for an angle \(\theta\):

   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
   \]
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
   \]
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
   \]

   These ratios help in determining the unknown sides or angles in right-angled triangles.

3. Reciprocal Trigonometric Functions:
   Alongside the primary trigonometric functions, there are their reciprocals:

   \[
   \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}
   \]
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}
   \]
   \[
   \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}
   \]

4. Pythagorean Identity:
   One of the key identities in trigonometry is derived from the Pythagorean theorem, which states that for a right-angled triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)):

   \[
   a^2 + b^2 = c^2
   \]

   Using the basic trigonometric functions, this can be expressed as:

   \[
   \sin^2(\theta) + \cos^2(\theta) = 1
   \]

5. Angle Sum and Difference Identities:
   Trigonometry also includes important identities for the sum and difference of angles, such as:

   \[
   \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)
   \]
   \[
   \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)
   \]

These foundational concepts of trigonometry are essential for solving a broad range of mathematical problems, extending into various applications in physics, engineering, and computer science. Trigonometry serves as a tool for understanding periodic phenomena, wave patterns, and even in the analysis of sound and light.