01 Key Concepts
Definition
A circle with radius exactly 1, centered at the origin (0,0) of the coordinate plane.
Coordinates on the Unit Circle
For any angle theta measured from the positive x-axis, the point on the unit circle is (cos(theta), sin(theta)).
Radians vs. Degrees
Radians measure angles using the circle's radius as a unit; 360° = 2*pi radians, so 180° = pi radians.
Key Angle Values
0°=0 radians, 90°=pi/2 radians, 180°=pi radians, 270°=3pi/2 radians, 360°=2*pi radians.
Signs in Each Quadrant
Quadrant I: both sin and cos positive. Quadrant II: sin positive, cos negative. Quadrant III: both negative. Quadrant IV: sin negative, cos positive.
02 Key Formulas
- 360° = 2*pi radians
- Point on unit circle at angle theta: (cos(theta), sin(theta))
03 Solved Examples
Example 1 Convert 90° to radians.
- Use the ratio: 180° = pi radians, so 1° = pi/180 radians.
- 90 * (pi/180) = pi/2.
Answer: pi/2 radians
Example 2 Find the coordinates on the unit circle at angle 0°.
- At 0°, cos(0°)=1 and sin(0°)=0.
Answer: (1, 0)
Example 3 In which quadrant is sin positive and cos negative?
- Check each quadrant's sign pattern for sin and cos.
- Quadrant II has positive sin and negative cos.
Answer: Quadrant II
04 Practice Questions
1Convert 180° to radians.
pi radians
2Convert 360° to radians.
2*pi radians
3Find the coordinates on the unit circle at 90°.
(0, 1)
4In which quadrant are both sin and cos negative?
Quadrant III
5Convert pi/2 radians to degrees.
90°
📄 Unit Circle — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.