Start with first-quadrant values
For 30°, 45° and 60°, the coordinate magnitudes come from 1/2, √2/2 and √3/2. The order swaps between sine and cosine.
Trigonometry · Free practice game
Unit Circle Practice that trains recall, not guessing.
Match standard angles and radians with their exact (cos θ, sin θ) coordinates. Ten questions, instant feedback, and a visual point on the circle every round.
The ray shows the angle. Choose its exact coordinate.
What are the exact coordinates at 30° (π/6)?
Remember: a unit-circle point is (cos θ, sin θ). Signs come from the quadrant.
Choose an answer to begin.
Round complete
Nice work.
What you are practicing
The unit circle turns angles into coordinates.
Every point on the unit circle has radius 1. At angle θ, the horizontal coordinate is cos θ and the vertical coordinate is sin θ, so the point is (cos θ, sin θ).
Then apply quadrant signs
Quadrant I is (+,+), II is (−,+), III is (−,−), and IV is (+,−). The reference-angle magnitudes stay the same.
Degrees and radians name the same ray
30° is π/6, 45° is π/4, 60° is π/3 and 90° is π/2. Practice both notations until the conversion feels automatic.
Core exact values
Memorize one quadrant, then use symmetry.
These five anchors generate the standard unit-circle coordinates. The practice game above expands them through all four quadrants.
| Degrees | Radians | cos θ | sin θ | Point |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | (1, 0) |
| 30° | π/6 | √3/2 | 1/2 | (√3/2, 1/2) |
| 45° | π/4 | √2/2 | √2/2 | (√2/2, √2/2) |
| 60° | π/3 | 1/2 | √3/2 | (1/2, √3/2) |
| 90° | π/2 | 0 | 1 | (0, 1) |
Cosine runs √4, √3, √2, √1, √0 over 2.
From 0° to 90°, cosine decreases as 1, √3/2, √2/2, 1/2, 0. Sine uses the same list in reverse.
Locate the quadrant before recalling the number.
At 150°, the reference angle is 30°. The magnitudes are √3/2 and 1/2, then Quadrant II makes x negative and y positive: (−√3/2, 1/2).
Common questions
Unit circle practice FAQ
What should I memorize on the unit circle?
Memorize the first-quadrant exact values for 0°, 30°, 45°, 60° and 90°, their radian equivalents, and the sign pattern by quadrant. Symmetry generates the rest.
Why is the point (cos θ, sin θ)?
On a circle of radius 1, cosine is the horizontal projection of the radius and sine is the vertical projection. Therefore the point reached by angle θ has coordinates (cos θ, sin θ).
How can I get faster without memorizing the whole circle at once?
Practice one direction at a time: first angle to coordinate, then coordinate to angle, then use mixed mode. Short repeated recall is more reliable than repeatedly rereading a completed chart.