Quadrant signs and standard unit-circle angles
Sine, cosine and tangent each have a fixed sign pattern in each of the four quadrants, and take exact values at the standard reference angles below.
| Quadrant | Angle range | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| I | 0°–90° | positive | positive | positive |
| II | 90°–180° | positive | negative | negative |
| III | 180°–270° | negative | negative | positive |
| IV | 270°–360° | negative | positive | negative |
- The reference angle is always the acute angle (0°–90°) between the terminal side and the x-axis; it is used with the quadrant's sign pattern to find exact trigonometric values for any angle from the standard 0°/30°/45°/60°/90° values.
- Tangent is undefined at 90° and 270° (and their coterminal equivalents), since cosine equals zero at those angles.
- Angles greater than 360° or negative angles are first reduced to a coterminal angle within [0°, 360°) before the quadrant and reference angle are determined.
What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin of a coordinate plane. For any angle θ measured counterclockwise from the positive x-axis, the point where the angle's terminal side crosses the circle has coordinates (cos θ, sin θ) — making the unit circle a visual and computational foundation for all trigonometric functions.
Because the circle's radius is exactly 1, sine and cosine values read directly as coordinates without any scaling. This calculator also reports the coterminal angle (the equivalent angle between 0° and 360°), the quadrant the terminal side falls in, and the reference angle (the acute angle to the nearest x-axis), which together explain the sign and magnitude of the trigonometric values.
How to use this unit circle calculator
- Enter the angle value.
- Select the angle unit: degrees or radians.
- Read sine, cosine and tangent for the angle, plus its quadrant, reference angle and coterminal angle (the equivalent angle within a single 0°–360° rotation).
The unit circle formulas
Because the unit circle has radius 1, a point's (x, y) coordinates are identical to (cos θ, sin θ) — no additional scaling by radius is needed, unlike a general circle.
Common mistakes
- Leaving the angle unit set to radians when entering a degree value, or vice versa, which produces a valid-looking but incorrect result.
- Assuming the reference angle equals the input angle for angles outside Quadrant I — the reference angle is always measured from the nearest x-axis, not from 0°.
- Expecting a numeric tangent value at 90° or 270°, where tangent is undefined because cosine is zero.
- Forgetting that negative angles and angles over 360° are equivalent to a coterminal angle within one full rotation (0°–360°).
常见问题
What are sin and cos at 225 degrees?
At 225°, sin(225°) = cos(225°) = −0.707107 (both equal to −√2/2). This is because 225° lies in Quadrant III, where the reference angle is 45° (225° − 180°) and both sine and cosine are negative.
What quadrant is 225 degrees in?
225° falls in Quadrant III (180°–270°), where both sine and cosine are negative and tangent is positive. Its reference angle is 225° − 180° = 45°.
What is a reference angle?
A reference angle is the acute angle, always between 0° and 90°, formed between the terminal side of an angle and the x-axis. For an angle of 225°, the reference angle is 45°, since 225° is 45° past the 180° mark.
What is a coterminal angle?
A coterminal angle is an angle that shares the same terminal side after being reduced to the range 0°–360° (or one full rotation). An angle of 585° has a coterminal angle of 585° − 360° = 225°, since both point to the same position on the circle.
Why is tan(90°) undefined on the unit circle?
Tangent equals sin θ / cos θ, and at 90° the point on the unit circle is (0, 1), meaning cos(90°) = 0. Dividing by zero is undefined, so tangent has no value at 90° (and at 270°, where cosine is also zero).
What are the exact sin and cos values at 30°, 45° and 60°?
sin(30°) = 0.5 and cos(30°) ≈ 0.866025; sin(45°) = cos(45°) ≈ 0.707107; sin(60°) ≈ 0.866025 and cos(60°) = 0.5. These are the most frequently memorized unit-circle values, all in Quadrant I.
参考文献
- Weisstein, Eric W. "Unit Circle" and "Reference Angle." MathWorld — A Wolfram Web Resource.
- Standard trigonometry textbook conventions (e.g. Larson, Trigonometry, Cengage Learning).