Mathematics: Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. Originating from the needs of astronomical measurements and navigation, trigonometry has now found applications across various fields such as physics, engineering, computer science, and even biology.

At its core, trigonometry explores the properties and functions related to the triangle’s angles and side lengths. Fundamental to this study are three primary trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are defined for an angle \(\theta\) in a right triangle with respect to the ratios of its sides:

• Sine (sin): The sine of an angle \(\theta\) is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, this is expressed as:
  \[
  \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
  \]

• Cosine (cos): The cosine of an angle \(\theta\) is the ratio of the length of the adjacent side to the length of the hypotenuse:
  \[
  \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
  \]

• Tangent (tan): The tangent of an angle \(\theta\) is the ratio of the length of the opposite side to the length of the adjacent side:
  \[
  \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
  \]

These functions are not only confined to right triangles but are also extended to the unit circle, enabling the definition of trigonometric functions for all angles, including those greater than 90 degrees. This extension leads to the concept of periodicity and the occurrence of trigonometric identities, which are equations involving trigonometric functions that hold true for all values of the occurring variables.

Some fundamental trigonometric identities include:

• Pythagorean Identity:
  \[
  \sin^2(\theta) + \cos^2(\theta) = 1
  \]

• Angle Sum and Difference Identities:
  \[
  \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)
  \]
  \[
  \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)
  \]

• Double Angle Identities:
  \[
  \sin(2\theta) = 2\sin(\theta)\cos(\theta)
  \]
  \[
  \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
  \]

Additionally, trigonometry includes the study of inverse trigonometric functions, which are used to determine angles given the value of a trigonometric function. These functions include \(\arcsin\), \(\arccos\), and \(\arctan\), which are crucial in solving triangles and modeling periodic phenomena.

Trigonometry also extends to complex numbers via Euler’s formula:
\[
e^{i\theta} = \cos(\theta) + i\sin(\theta)
\]
where \(i\) is the imaginary unit.

Overall, trigonometry is an essential part of the mathematical toolkit that enables the solution of practical problems involving angular relationships and periodic processes. Its principles and methods provide foundational knowledge that is instrumental in both theoretical and applied sciences.