Angle in Standard Position
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An angle in the coordinate plane is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.
Vertex at (0, 0) · Initial side on the positive x-axis
Counterclockwise rotation gives a positive angle. Clockwise rotation gives a negative angle.
The parts of an angle in standard position
Think of the angle as a ray that starts on the positive x-axis and rotates around the origin.
- Vertex: the origin, the point the ray rotates around.
- Initial side: the starting position of the ray. In standard position it always lies along the positive x-axis.
- Terminal side: the position of the ray after it has rotated.
The measure of the angle is the amount and direction of rotation needed to move from the initial side to the terminal side.
Positive and negative angles
The direction of rotation decides the sign of the angle:
- A counterclockwise rotation creates a positive angle, such as 120°.
- A clockwise rotation creates a negative angle, such as −60°.
A ray can also rotate more than one full turn. One full turn is 360°, so an angle of 400° makes one complete turn and then 40° more.
See it on the coordinate plane
In the figure below, the initial side (red) lies along the positive x-axis, and the curved arrow shows the rotation to the terminal side (blue).
The next two figures show the same terminal side reached in two different ways. Rotating counterclockwise gives a positive angle. Rotating clockwise, the long way around, gives a negative angle.
Angles that share the same terminal side, like these two, are called coterminal angles.
Which quadrant is the angle in?
An angle in standard position "lies in" the quadrant where its terminal side lands. For angles between 0° and 360°:
When the terminal side lands exactly on an axis (0°, 90°, 180°, 270° or 360°), the angle is not in any quadrant. These are called quadrantal angles.
Coterminal angles
Adding or subtracting a full turn (360°) brings the terminal side back to the same place. So every angle has infinitely many coterminal angles:
Coterminal angles of θ: θ + 360°·k, where k is any integer
In radians, one full turn is 2π, so the coterminal angles are θ + 2πk. See radian measure.
Examples
| Angle | Rotation | Terminal side |
|---|---|---|
| 150° | Counterclockwise 150° | Quadrant II |
| −60° | Clockwise 60° | Quadrant IV (coterminal with 300°) |
| 400° | One full turn, then 40° more | Quadrant I (coterminal with 40°) |
| −235° | Clockwise 235° | Quadrant II (coterminal with 125°) |
| 270° | Counterclockwise 270° | Negative y-axis (quadrantal) |
Tip for negative angles: add 360° until the result is between 0° and 360°. For −235°: −235° + 360° = 125°, which is in Quadrant II.
Practice problems
Pick an answer for each question. You'll see right away whether it's correct, with an explanation.
1. In which quadrant does an angle of 200° lie?
200° is between 180° and 270°, so it lies in Quadrant III.
2. In which quadrant does an angle of −120° lie?
A negative angle rotates clockwise. Add 360° to find the matching positive angle: −120° + 360° = 240°, which is between 180° and 270°. The angle lies in Quadrant III.
3. Which pair of angles is coterminal with 75°?
Add or subtract a full turn of 360°: 75° + 360° = 435° and 75° − 360° = −285°.
4. In which quadrant does an angle of 3π/4 radians lie?
Convert to degrees: 3π/4 × (180°/π) = 135°. That is between 90° and 180°, so the angle lies in Quadrant II.
5. Where is the terminal side of a 720° angle?
720° = 2 × 360°, which is exactly two full turns. The terminal side ends where it started, on the positive x-axis, so 720° is a quadrantal angle coterminal with 0°.