Coterminal angles consistently occupy the same quadrant on the coordinate plane because they share an identical terminal side when in standard position. Despite having different angular measurements, these angles begin at the same initial side (the positive x-axis) and conclude at the exact same position, meaning their terminal sides coincide.
Understanding Coterminal Angles
Coterminal angles are defined as angles that have the same initial side and share the same terminal side. Although their numerical values differ, they effectively point in the same direction from the origin. These angles are always positioned in standard form, where their vertices are identical (at the origin), and their initial side lies along the positive x-axis. Because their terminal sides are shared, they inherently fall within the same region or quadrant of the coordinate system.
Why Coterminal Angles Share a Quadrant
The fundamental reason coterminal angles share a quadrant is their common terminal side. An angle's quadrant is determined by where its terminal side lands after rotation. Since coterminal angles always finish their rotation at the exact same terminal position, they must, by definition, reside in the same quadrant. For instance, if an angle's terminal side falls in Quadrant I, all its coterminal angles will also have their terminal sides in Quadrant I.
The Four Quadrants
The coordinate plane is divided into four quadrants, defined by the signs of the x and y coordinates. Understanding these divisions is crucial for identifying an angle's quadrant.
| Quadrant | Angle Range (Degrees) | Angle Range (Radians) | (x, y) Coordinates | Description |
|---|---|---|---|---|
| I | $0^\circ < \theta < 90^\circ$ | $0 < \theta < \pi/2$ | (+, +) | Top-right section |
| II | $90^\circ < \theta < 180^\circ$ | $\pi/2 < \theta < \pi$ | (-, +) | Top-left section |
| III | $180^\circ < \theta < 270^\circ$ | $\pi < \theta < 3\pi/2$ | (-, -) | Bottom-left section |
| IV | $270^\circ < \theta < 360^\circ$ | $3\pi/2 < \theta < 2\pi$ | (+, -) | Bottom-right section |
Note: Angles whose terminal sides lie exactly on an axis (e.g., $0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ$) are considered quadrantal angles and do not strictly belong to a quadrant but rather separate them. For more details on quadrants, you can refer to resources on the Cartesian coordinate system.
Examples of Coterminal Angles and Their Quadrants
Here are a few examples illustrating how coterminal angles always share the same quadrant:
- Quadrant I:
- An angle of $45^\circ$ lies in Quadrant I.
- Coterminal angles like $405^\circ$ ($45^\circ + 360^\circ$) and $-315^\circ$ ($45^\circ - 360^\circ$) also have their terminal sides in Quadrant I.
- Quadrant II:
- An angle of $130^\circ$ lies in Quadrant II.
- Coterminal angles such as $490^\circ$ ($130^\circ + 360^\circ$) and $-230^\circ$ ($130^\circ - 360^\circ$) also terminate in Quadrant II.
- Quadrant III:
- An angle of $210^\circ$ lies in Quadrant III.
- Coterminal angles like $570^\circ$ ($210^\circ + 360^\circ$) and $-150^\circ$ ($210^\circ - 360^\circ$) also reside in Quadrant III.
- Quadrant IV:
- An angle of $300^\circ$ lies in Quadrant IV.
- Coterminal angles such as $660^\circ$ ($300^\circ + 360^\circ$) and $-60^\circ$ ($300^\circ - 360^\circ$) also terminate in Quadrant IV.
Calculating Coterminal Angles
To find coterminal angles, you simply add or subtract multiples of a full rotation ($360^\circ$ or $2\pi$ radians) from the given angle.
- In degrees: $\theta_{\text{coterminal}} = \theta \pm 360^\circ \times n$
- In radians: $\theta_{\text{coterminal}} = \theta \pm 2\pi \times n$
Where $\theta$ is the original angle and $n$ is any positive integer (1, 2, 3, ...). Each rotation, whether positive (counter-clockwise) or negative (clockwise), brings the terminal side back to its original position, hence maintaining the same quadrant.
Practical Insights
Understanding coterminal angles is vital in trigonometry. Since they share the same terminal side, all trigonometric functions (sine, cosine, tangent, etc.) for coterminal angles will yield the same values. For example, $\sin(30^\circ) = \sin(390^\circ) = \sin(-330^\circ)$. This property simplifies calculations and is fundamental in studying periodic functions.
