docs: enhance Trigonometry 01 with comprehensive improvements and MathWorld links

docs/src/Trigonometry/01 Angles & Measurements.md

@@ -4,122 +4,162 @@ Understanding angles and their measurement is fundamental to trigonometry and ge
 
 ## Angle Definition
 
-An angle is formed by two rays (called sides) that share a common endpoint (called the vertex).
+An [angle](https://mathworld.wolfram.com/Angle.html) is formed by two rays (called sides) that share a common endpoint (called the vertex).
 
 ## Angle Measurement Systems
 
 ### Degrees
+
+A [degree](https://mathworld.wolfram.com/Degree.html) is a unit of angular measurement where a full rotation around a circle is divided into 360 equal parts.
+
+#### Degree Definition
+
+One degree (1°) is defined as 1/360 of a complete rotation around a circle.
+
+#### Key Values
+
 - **Full rotation:** 360°
 - **Right angle:** 90°
 - **Straight angle:** 180°
 - **Common angles:** 30°, 45°, 60°, 90°
 
 ### Radians
-Radians provide a natural unit for angle measurement based on arc length.
 
-#### Definition
-One radian is the angle subtended by an arc equal in length to the radius of the circle.
+[Radians](https://mathworld.wolfram.com/Radian.html) provide a natural unit for angle measurement based on arc length.
+
+#### Radian Definition
+
+One radian is the angle subtended by an [arc](https://mathworld.wolfram.com/Arc.html) equal in length to the radius of the circle.
+
+**Note:** An arc is a portion of the circumference of a circle - the curved line that forms part of the circle's edge.
 
 #### Key Relationships
-- $2\pi$ radians = 360°
-- $\pi$ radians = 180°
-- $\frac{\pi}{2}$ radians = 90°
-- $\frac{\pi}{3}$ radians = 60°
-- $\frac{\pi}{4}$ radians = 45°
-- $\frac{\pi}{6}$ radians = 30°
+
+$$\begin{array}{c|c}
+\text{Radians} & \text{Degrees} \\
+\hline
+2\pi & 360° \\
+\hline
+\pi & 180° \\
+\hline
+\frac{\pi}{2} & 90° \\
+\hline
+\frac{\pi}{3} & 60° \\
+\hline
+\frac{\pi}{4} & 45° \\
+\hline
+\frac{\pi}{6} & 30° \\
+\hline
+\end{array}$$
 
 #### Conversion Formulas
+
 - **Degrees to radians:** $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
 - **Radians to degrees:** $\text{degrees} = \text{radians} \times \frac{180}{\pi}$
 
 ### Gradians (Gons)
+
+A [gradian](https://mathworld.wolfram.com/Gradian.html) (also called a gon or grad) is a unit of angular measurement where a full rotation around a circle is divided into 400 equal parts.
+
+#### Gradian Definition
+
+One gradian (1ᵍ) is defined as 1/400 of a complete rotation around a circle.
+
+#### Gradian Values
+
 - **Full rotation:** 400 gradians
 - **Right angle:** 100 gradians
+
+#### Usage Notes
+
 - Less commonly used in modern applications
+- Primarily used in surveying and some European engineering contexts
 
 ## Types of Angles
 
 ### By Measure
+
 - **Acute:** $0° < \theta < 90°$ (or $0 < \theta < \frac{\pi}{2}$)
 - **Right:** $\theta = 90°$ (or $\theta = \frac{\pi}{2}$)
 - **Obtuse:** $90° < \theta < 180°$ (or $\frac{\pi}{2} < \theta < \pi$)
 - **Straight:** $\theta = 180°$ (or $\theta = \pi$)
 - **Reflex:** $180° < \theta < 360°$ (or $\pi < \theta < 2\pi$)
 
 ### By Position
-- **Standard position:** Vertex at origin, initial side on positive x-axis
-- **Coterminal angles:** Angles that differ by multiples of $360°$ (or $2\pi$ radians)
+
+- **[Standard position](https://mathworld.wolfram.com/AngleStandardPosition.html):** Vertex at origin, initial side on positive x-axis
+- **[Coterminal angles](https://mathworld.wolfram.com/CoterminalAngle.html):** Angles that differ by multiples of $360°$ (or $2\pi$ radians)
 - **Reference angle:** Acute angle between terminal side and x-axis
+- **Negative angles:** Angles measured clockwise from the positive x-axis
+
+## Arc Length and Sector Area
 
-## Angle Relationships
+Understanding how angles relate to circular measurements is fundamental for applications ranging from engineering and physics to computer graphics and navigation. When we measure angles, we often need to calculate the actual lengths and areas they create on circles.
 
-### Complementary Angles
-Two angles whose measures sum to 90°:
-- $\sin \theta = \cos(90° - \theta)$
-- $\cos \theta = \sin(90° - \theta)$
-- $\tan \theta = \cot(90° - \theta)$
+### Arc Length
 
-### Supplementary Angles
-Two angles whose measures sum to 180°:
-- $\sin(180° - \theta) = \sin \theta$
-- $\cos(180° - \theta) = -\cos \theta$
-- $\tan(180° - \theta) = -\tan \theta$
+#### Arc Length Definition and Importance
 
-## Arc Length and Sector Area
+[Arc length](https://mathworld.wolfram.com/ArcLength.html) is the distance along the curved path of a circle between two points. This measurement is crucial because:
+
+- **Engineering applications:** Calculating belt lengths, gear rotations, and mechanical movements
+- **Navigation:** Determining distances along curved paths on Earth's surface
+- **Physics:** Understanding rotational motion and wave phenomena
+
+#### Arc Length Formula and Usage
 
-### Arc Length
 For a circle with radius $r$ and central angle $\theta$ (in radians):
+
 $$s = r\theta$$
 
+**Key insight:** This formula only works when the angle is measured in radians, which is why radians are considered the "natural" unit for angular measurement.
+
 ### Sector Area
+
+#### Sector Area Definition and Importance
+
+A [sector](https://mathworld.wolfram.com/CircularSector.html) is a "slice" of a circle, like a piece of pie. Sector area calculations are essential for:
+
+- **Construction and design:** Calculating areas for circular segments in architecture
+- **Statistics:** Creating pie charts and circular data visualizations  
+- **Agriculture:** Determining irrigation coverage areas
+- **Astronomy:** Calculating field-of-view areas for telescopes
+
+#### Sector Area Formula and Usage
+
 $$A = \frac{1}{2}r^2\theta$$
 
-### Angular Velocity
-Rate of change of angle with respect to time:
-$$\omega = \frac{d\theta}{dt}$$
+**Note:** Like arc length, this formula requires the angle $\theta$ to be in radians. The factor of $\frac{1}{2}$ comes from the relationship between the sector area and the full circle area $\pi r^2$.
 
-Linear velocity relates to angular velocity by: $v = r\omega$
+### Angular Velocity
 
-## Special Angle Values
+#### Angular Velocity Definition and Physical Significance
 
-### Exact Values Table
+[Angular velocity](https://mathworld.wolfram.com/AngularVelocity.html) measures how fast an angle changes over time, which is fundamental in describing rotational motion:
 
-| Angle (°) | Angle (rad) | sin | cos | tan |
-|-----------|-------------|-----|-----|-----|
-| 0° | 0 | 0 | 1 | 0 |
-| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
-| 45° | π/4 | √2/2 | √2/2 | 1 |
-| 60° | π/3 | √3/2 | 1/2 | √3 |
-| 90° | π/2 | 1 | 0 | undefined |
+$$\omega = \frac{d\theta}{dt}$$
 
-### Memory Aids
-- **30-60-90 triangle:** Sides in ratio $1 : \sqrt{3} : 2$
-- **45-45-90 triangle:** Sides in ratio $1 : 1 : \sqrt{2}$
+#### Relationship to Linear Motion
 
-## Periodic Nature of Angles
+The connection between rotational and linear motion is given by:
 
-### Periodicity
-Trigonometric functions repeat their values at regular intervals:
-- $\sin(\theta + 2\pi) = \sin \theta$
-- $\cos(\theta + 2\pi) = \cos \theta$
-- $\tan(\theta + \pi) = \tan \theta$
+$$v = r\omega$$
 
-### Principal Values
-- For inverse trig functions, we define principal value ranges:
-  - $\arcsin$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$
-  - $\arccos$: $[0, \pi]$
-  - $\arctan$: $(-\frac{\pi}{2}, \frac{\pi}{2})$
+Where $v$ is the linear velocity of a point at distance $r$ from the rotation center. This relationship explains why points farther from the center of rotation move faster linearly while maintaining the same angular velocity.
 
 ## Applications
 
 ### Navigation
+
 - **Bearing:** Direction measured clockwise from north
 - **Azimuth:** Angle measured from a reference direction
 
 ### Engineering
+
 - **Phase angles:** In electrical engineering for AC circuits
 - **Rotation:** Describing rotational motion in mechanics
 
 ### Computer Graphics
+
 - **Rotation matrices:** Using angles to rotate objects
-- **Animation:** Smooth transitions using trigonometric functions
+- **Animation:** Smooth transitions using mathematical functions for rotation and movement

docs/src/Trigonometry/02 Trigonometric Functions.md

@@ -65,3 +65,59 @@ Hyperbolic functions have many similarities to trigonometric functions:
 - **Hyperbolic identity:** $\cosh^2(x) - \sinh^2(x) = 1$
 
 The key difference is the sign in the fundamental identity, which reflects the difference between the unit circle ($x^2 + y^2 = 1$) and the unit hyperbola ($x^2 - y^2 = 1$).
+
+## TODO: Content Moved from Document 01
+
+The following sections were moved from "01 Angles & Measurements.md" because they reference trigonometric functions that need to be defined first. These sections need to be properly integrated into this document or moved to their appropriate locations:
+
+### Angle Relationships
+
+#### Complementary Angles
+
+Two angles whose measures sum to 90°:
+
+- $\sin \theta = \cos(90° - \theta)$
+- $\cos \theta = \sin(90° - \theta)$
+- $\tan \theta = \cot(90° - \theta)$
+
+#### Supplementary Angles
+
+Two angles whose measures sum to 180°:
+
+- $\sin(180° - \theta) = \sin \theta$
+- $\cos(180° - \theta) = -\cos \theta$
+- $\tan(180° - \theta) = -\tan \theta$
+
+### Special Angle Values
+
+#### Exact Values Table
+
+| Angle (°) | Angle (rad) | sin | cos | tan |
+|-----------|-------------|-----|-----|-----|
+| 0° | 0 | 0 | 1 | 0 |
+| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
+| 45° | π/4 | √2/2 | √2/2 | 1 |
+| 60° | π/3 | √3/2 | 1/2 | √3 |
+| 90° | π/2 | 1 | 0 | undefined |
+
+#### Memory Aids
+
+- **30-60-90 triangle:** Sides in ratio $1 : \sqrt{3} : 2$
+- **45-45-90 triangle:** Sides in ratio $1 : 1 : \sqrt{2}$
+
+### Periodic Nature of Angles
+
+#### Periodicity
+
+Trigonometric functions repeat their values at regular intervals:
+
+- $\sin(\theta + 2\pi) = \sin \theta$
+- $\cos(\theta + 2\pi) = \cos \theta$
+- $\tan(\theta + \pi) = \tan \theta$
+
+#### Principal Values
+
+- For inverse trig functions, we define principal value ranges:
+  - $\arcsin$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$
+  - $\arccos$: $[0, \pi]$
+  - $\arctan$: $(-\frac{\pi}{2}, \frac{\pi}{2})$