Unit Circle
Circle of radius 1; point at angle θ is (cosθ, sinθ).
Key idea
Circle of radius 1; point at angle θ is (cosθ, sinθ).
In depth
The unit circle has radius 1 centred at the origin. The point at angle θ is (cos θ, sin θ). This extends sine and cosine to all angles, including negative and above 90°, and builds in x² + y² = 1.
Worked example
At 120°, the point is (−1/2, √3/2), so cos 120° = −1/2 and sin 120° = √3/2.
Formula
Real-world uses
- Modelling rotation in games and robotics
- Explaining the shape of sine and cosine waves
Common mistake
Assigning cos to the y-coordinate.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Unit Circle?
Circle of radius 1; point at angle θ is (cosθ, sinθ). It is a topic at the intermediate level in Trigonometry.
What is the formula for Unit Circle?
The key formula is x² + y² = 1.
Where is Unit Circle used in real life?
Real-world examples: Modelling rotation in games and robotics; Explaining the shape of sine and cosine waves.
What should I learn before Unit Circle?
Make sure you are comfortable with Special Angle Values, the previous topic in Trigonometry.
What is a common mistake with Unit Circle?
Assigning cos to the y-coordinate.
How can I practise Unit Circle?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Unit Circle?
Move on to Reference Angles next.
Related intermediate topics in Trigonometry
- Radians – Angle measured by arc length on a unit circle; 2π rad = 360°.
- Converting Degrees and Radians – Multiply by π/180 or 180/π.
- Sine Ratio – Opposite over hypotenuse.
- Cosine Ratio – Adjacent over hypotenuse.
- Tangent Ratio – Opposite over adjacent.
- Reciprocal Trig Functions – Cosecant, secant and cotangent.