The Unit Circle
A circle of radius 1 where cos θ and sin θ are the coordinates.
About this topic
The unit circle has radius 1 and is centred at the origin. A point on it at angle θ has coordinates (cos θ, sin θ), which defines sine and cosine for all angles, including those beyond 90° and negative angles. The equation is x² + y² = 1, which gives sin²θ + cos²θ = 1.
To use it, measure θ counterclockwise from the positive x-axis for positive angles and clockwise for negative ones. The x-coordinate gives the cosine and the y-coordinate gives the sine, and tan θ is y/x, the slope of the radius. Going past 360° simply wraps around again, which is why trig functions are periodic.
Example: at 90° the point is (0, 1), so cos 90° = 0 and sin 90° = 1. At 210° the point lies in the third quadrant with both coordinates negative, namely (−√3/2, −1/2). Tracking a point around the circle and plotting its height against the angle produces the sine wave.
The unit circle is the foundation for waves, rotations, alternating current and sound. Students often memorise it as a chart, but it is easier to learn the first quadrant and use symmetry. Remember the quadrant signs: all positive in I, sine in II, tangent in III, cosine in IV.
Key ideas
- Point is (cos θ, sin θ)
- Signs follow quadrants
- Special angles have exact values
Quick check
Use cases: where The Unit Circle helps
- Homework: review the key idea behind The Unit Circle before you start a problem set.
- Exam revision: use the quick check to test whether you can recall The Unit Circle without notes.
- Visual learning: play with the interactive lab to see how The Unit Circle changes when you adjust its inputs.
- Teaching a friend: explain The Unit Circle in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is The Unit Circle?
A circle of radius 1 where cos θ and sin θ are the coordinates.
What will I learn in The Unit Circle?
Point is (cos θ, sin θ). Signs follow quadrants. Special angles have exact values.
Is The Unit Circle part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the The Unit Circle lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study The Unit Circle?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.