The unit circle has center at the origin and radius 1. Starting from the point (1, 0), an angle moves around the circle counterclockwise when positive and clockwise when negative. Its final point connects angle measurement with sine and cosine.
How do radians measure an angle?
A radian measures an angle by dividing its corresponding arc length s by the circle’s radius r. A full turn is 360° or twice π radians. On the unit circle, the numerical arc length equals the angle in radians because the radius is one.
To convert degrees to radians, multiply by π and divide by 180. Reverse that factor to convert back. In degree-minute-second notation, a degree contains 60 minutes and a minute contains 60 seconds; these angular units are different from time units.
Coordinates determine the trigonometric values
At the final point P, the horizontal coordinate is cosine and the vertical coordinate is sine. Their squared coordinates add to one. Tangent is the vertical coordinate divided by the horizontal one, so it is undefined when the horizontal coordinate is zero.
Quadrants and signs
Quadrant I lies strictly between 0° and 90° and has both coordinates positive. In quadrant II, cosine is negative and sine positive. Both are negative in quadrant III. In quadrant IV, cosine is positive and sine negative. Points on the axes do not belong to a quadrant’s interior.
Worked example: locate 150°
Convert 150° to radians, then use its 30° reference angle. The point is in quadrant II, so its horizontal coordinate is negative and its vertical coordinate positive. This gives the exact coordinates below.
Negative angles and extra turns
Adding or subtracting a full turn changes the travelled angle but leaves the final point unchanged. For example, −210° and 150° finish at the same point. Keep the original angle when describing movement; use an equivalent angle to locate its position.
Explore a full turn
Move the slider or use its arrow keys. The point and its coordinates update together; the text output gives the same numerical information as the graph.
For the algebraic equation behind this circle, see the equation of a circle.
