Equivalent angles θ + n × 360°: finding a positive, a negative and the smallest positive measure of an angle
Adding or subtracting a full turn does not move the terminal side. So θ and θ + n × 360° (n any whole number, positive, negative or zero) are equivalent. To find the smallest positive measure, add or subtract multiples of 360° until the result is from 0° up to (but not including) 360°; with a big angle divide by 360 and keep the remainder. For a negative angle add 360° until it becomes positive. Equivalent angles have exactly the same position in the plane, so they have the same quadrant and later the same trigonometric ratios. Always check the answer is in [0°, 360°[.
| Angle | Add or subtract | Smallest positive |
|---|---|---|
| 765° | − 2 × 360° | 45° |
| −400° | + 2 × 360° | 320° |
| −1000° | + 3 × 360° | 80° |
| 2000° | − 5 × 360° | 200° |
Wheel game. Mark a bicycle wheel valve at 0° (pointing right). Roll the wheel forward by 2 turns and 50° more. Predict where the valve is, then say the angle as a number over 360°.