Trigonometry: Directed Angles, Standard Position, and Degree-Radian Conversion
A directed angle expands geometric intuition by incorporating the direction of rotation. An angle is in standard position when its vertex is at the origin (0, 0) and its initial side coincides with the positive x-axis. Rotation in a counter-clockwise direction produces a positive measure, whereas clockwise rotation yields a negative measure. We examine coterminal angles (θ ± 360° n) and establish the radian measure: the central angle subtending an arc whose length equals the radius (s = r · θ rad). Precise conversion between sexagesimal degrees and circular radians (D° / 180° = θ rad / π ) completes this foundation.
1. Directed Angle in Standard Position
Core geometric principles:
- Standard Position:
1. Vertex at origin (0, 0).
2. Initial side along positive x-axis (OX). - Sign of Rotation:
• Counter-clockwise = Positive (+).
• Clockwise = Negative (-). - Coterminal Angles: Directed angles sharing the same terminal side: θ + 360°k (where k ∈ ℤ).
2. Radian Measure & Arc Length
Formulae and conversion:
- Radian Definition: θrad = s / r ⇒ s = r · θrad (where s is arc length, r is radius).
- Golden Conversion Formula:
• D° / 180° = θrad / π
• To radians: θrad = (D° · π) / 180°.
• To degrees: D° = (θrad · 180°) / π.
Worked Application Examples:
Solution: Convert to radians: θrad = (120 × π) / 180 = 2π / 3. Arc length s = r · θrad = 9 × (2π / 3) = 6π ≈ 18.85 cm.
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