3.4 The Trigonometric Circle - Arcs and Sectors
1. Concept of the Trigonometric Circle
The trigonometric circle geometrically maps angle values onto a 360-degree continuous boundary. Starting securely from the positive x-axis and rotating anticlockwise maps positive orientations ($0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ$). Rotating physically clockwise dictates mathematically negative values ($-90^\circ$ acts identically to $270^\circ$).
Periodic properties allow angles extending far past $360^\circ$ to functionally overlap points on the initial cycle. The general position on the boundary is maintained exactly through the expression $\theta \pm 360^\circ k$ (where $k$ is any integer multiple).
2. Degrees and Radians
Radians act as an alternative unit of measurement correlating geometrically directly to the perimeter arc length traced along a pure unit circle ($r=1$). A full $360^\circ$ revolution traces precisely the complete circumference distance $2\pi$.
| Degrees | $0^\circ$ | $90^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|
| Radians | $0$ | $\frac{\pi}{2}$ | $\pi$ | $\frac{3\pi}{2}$ | $2\pi$ |
Conversion Formula: The proportional ratio mapping degrees to radians is $\frac{\text{Degrees}}{180^\circ} = \frac{\text{Radians}}{\pi}$.
EXAMPLE 1 & 2 (Unit Conversions)
$\theta_1 = 30^\circ \Rightarrow \frac{30^\circ}{x} = \frac{180^\circ}{\pi} \Rightarrow x = \mathbf{\frac{\pi}{6}\text{ rad}}$.
$\theta_2 = 80^\circ \Rightarrow \frac{80^\circ}{x} = \frac{180^\circ}{\pi} \Rightarrow x = \mathbf{\frac{4\pi}{9}\text{ rad}}$.
$\theta_3 = 27^\circ \Rightarrow x = \mathbf{0.471\text{ rad}}$.
$\theta_1 = \frac{\pi}{3} \Rightarrow \frac{x}{\pi/3} = \frac{180^\circ}{\pi} \Rightarrow x = \mathbf{60^\circ}$.
$\theta_2 = \frac{4\pi}{9} \Rightarrow \frac{x}{4\pi/9} = \frac{180^\circ}{\pi} \Rightarrow x = \mathbf{80^\circ}$.
$\theta_3 = 2 \text{ rad} \Rightarrow \frac{x}{2} = \frac{180^\circ}{\pi} \Rightarrow x = \frac{360}{\pi} = \mathbf{114.6^\circ}$.
3. Length of Arcs and Area of Sectors
Given a circle possessing radius $r$ and a central interior angle denoted as $\theta$ (must strictly be measured in radians):
- Length of Arc ($L$): $L = r\theta$
- Area of Sector ($A$): $A = \frac{1}{2}r^2\theta$
Note: The area isolated structurally between a chord line segment AB and the circular arc boundary AB (a geometric segment) is given by $A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$.
EXAMPLE 3
Consider a bounded sector characterized by radius $r = 5\text{m}$ and angle $\theta = 0.6\text{ rad}$.
EXAMPLE 4
Given geometric parameters $r = 3\text{cm}$ and $\theta = 30^\circ$, evaluate dimensional features.
$AB^2 = 3^2 + 3^2 - 2(3)(3)\cos\left(\frac{\pi}{6}\right) \Rightarrow AB \approx \mathbf{1.55\text{cm}}$.