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	$0$	$\dfrac{\pi}{2}$	$\pi$	$\dfrac{3\pi}{2}$	$2\pi$

Conversion Formula: The proportional ratio mapping degrees to radians is:

$$\dfrac{\text{Degrees}}{180^\circ} = \dfrac{\text{Radians}}{\pi}$$

EXAMPLE 1 (Transform into Radians)

$\theta_1 = 30^\circ \implies \dfrac{30^\circ}{x} = \dfrac{180^\circ}{\pi} \implies x = \mathbf{\dfrac{\pi}{6}\text{ rad}}$

$\theta_2 = 80^\circ \implies \dfrac{80^\circ}{x} = \dfrac{180^\circ}{\pi} \implies x = \mathbf{\dfrac{4\pi}{9}\text{ rad}}$

$\theta_3 = 27^\circ \implies x = 27 \times \dfrac{\pi}{180} \implies x \approx \mathbf{0.471\text{ rad}}$

EXAMPLE 2 (Transform into Degrees)

$\theta_1 = \dfrac{\pi}{3} \implies \dfrac{x}{\pi/3} = \dfrac{180^\circ}{\pi} \implies x = \mathbf{60^\circ}$

$\theta_2 = \dfrac{4\pi}{9} \implies \dfrac{x}{4\pi/9} = \dfrac{180^\circ}{\pi} \implies x = \mathbf{80^\circ}$

$\theta_3 = 2 \text{ rad} \implies \dfrac{x}{2} = \dfrac{180^\circ}{\pi} \implies x = \dfrac{360^\circ}{\pi} \approx \mathbf{114.6^\circ}$

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 ($A$): $A = \dfrac{1}{2}r^2\theta$

Note: The area isolated structurally between a chord line segment $AB$ and the circular arc boundary $AB$ (a geometric segment) is derived by subtracting the triangle area from the sector area:

$$A_{\text{segment}} = \dfrac{1}{2}r^2(\theta - \sin\theta)$$

EXAMPLE 3

Consider a bounded sector characterized by radius $r = 5\text{m}$ and angle $\theta = 0.6\text{ rad}$.

Length of circular arc:

$L = 5 \times 0.6 = \mathbf{3\text{m}}$

Area of bound sector:

$A = \dfrac{1}{2}(5^2)(0.6) = \mathbf{7.5\text{m}^2}$

Perimeter surrounding sector:

$P = L + r + r = 3 + 5 + 5 = \mathbf{13\text{m}}$

EXAMPLE 4

Given geometric parameters $r = 3\text{cm}$ and $\theta = 30^\circ$, evaluate dimensional features.

First, structurally convert degrees to radians:

$\theta = 30^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{\pi}{6}\text{ rad}$

(a) Arc length AB:

$L = 3\left(\dfrac{\pi}{6}\right) = \dfrac{\pi}{2} \approx \mathbf{1.57\text{cm}}$

(b) Sector Area OAB:

$A_{\text{sec}} = \dfrac{1}{2}(3^2)\left(\dfrac{\pi}{6}\right) = \dfrac{9\pi}{12} = \dfrac{3\pi}{4} \approx \mathbf{2.36\text{cm}^2}$

(c) Straight distance AB (chord length): Utilizing Cosine Rule evaluating the full triangle:

$$\begin{aligned} AB^2 &= 3^2 + 3^2 - 2(3)(3)\cos\left(\dfrac{\pi}{6}\right) \\ AB^2 &= 18 - 18\left(\dfrac{\sqrt{3}}{2}\right) \\ AB^2 &= 18 - 9\sqrt{3} \implies AB \approx \mathbf{1.55\text{cm}} \end{aligned}$$

(d) Area of triangle OAB:

$A_{\text{tri}} = \dfrac{1}{2}(3)(3)\sin\left(\dfrac{\pi}{6}\right) = \dfrac{9}{2} \times \dfrac{1}{2} = \dfrac{9}{4} = \mathbf{2.25\text{cm}^2}$