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Quick examples
How to use this calculator
- Select the coterminal angle operation you need.
- Enter the angle and choose degrees or radians.
- Use pi notation like pi/6, 5*pi/3, or -7*pi/4 for radians.
- Press Calculate to view the result, formula, steps, table, and visual preview.
- Use Download CSV or Print / PDF for saving the result.
Formula used
Coterminal angles are angles that end on the same terminal side in standard position. In degrees, the family is θ + 360°k. In radians, the family is θ + 2πk. The integer k can be positive, negative, or zero.
The standard coterminal angle is usually placed in the interval 0° ≤ θ < 360° or 0 ≤ θ < 2π. A principal signed angle may also use -180° < θ ≤ 180°. Both forms describe the same ray, but the chosen interval changes the displayed answer.
Understanding coterminal angles
Coterminal angles are central to trigonometry because sine, cosine, tangent, and the reciprocal ratios repeat after each full revolution. A 45° angle, a 405° angle, and a -315° angle point in the same direction. They are different rotations, but they share one terminal side.
Why full turns matter
A full turn is 360° or 2π radians. Adding a full turn moves around the circle and returns to the same place. Subtracting a full turn does the same thing in the opposite direction. That is why the integer k appears in coterminal formulas.
Degrees and radians
Degree problems often look easier because 360 is familiar. Radian problems use 2π instead. The idea is identical. For example, π/3 and 7π/3 are coterminal because 7π/3 − π/3 = 2π.
Quadrants and reference angles
The standard coterminal angle helps identify the quadrant. Once the quadrant is known, the reference angle gives the acute angle to the x-axis. This is useful for sign rules and exact trigonometric values.
Practical uses
Coterminal angles appear in unit-circle work, graphing trigonometric functions, navigation headings, rotations, vectors, complex numbers, and periodic equations. A calculator that shows the family and the reduced angle helps avoid sign and quadrant mistakes.
FAQs
What is a coterminal angle?
It is an angle that shares the same terminal side as another angle in standard position.
How do I find coterminal degrees?
Add or subtract 360° as many times as needed.
How do I find coterminal radians?
Add or subtract 2π as many times as needed.
Can coterminal angles be negative?
Yes. A negative coterminal angle rotates clockwise to the same terminal side.
What is the least positive coterminal angle?
It is the smallest angle greater than zero that shares the same terminal side.
Are 30° and 390° coterminal?
Yes. Their difference is 360°, which is one complete turn.
Are π/6 and 13π/6 coterminal?
Yes. Their difference is 2π.
Why do trig values repeat?
The unit-circle coordinates repeat after every full rotation.
Does the quadrant change for coterminal angles?
No. Coterminal angles share the same terminal side, so their standard quadrant is the same.
What interval is standard?
Common standard intervals are [0°,360°) and [0,2π).
What is a principal angle?
It is a signed coterminal angle with the smallest rotation size, often in (-180°,180°].
Can I type pi values?
Yes. Use values like pi/3, 5*pi/6, or -7*pi/4.
What does k mean?
k is any integer representing the number of complete rotations.
Can I export the answer?
Yes. Use the CSV button or Print / PDF button.