The Unit Circle
The unit circle is one of the most powerful tools in trigonometry. It connects angles to coordinates in a beautifully simple way.
A circle with radius 1 centred at the origin in the Cartesian plane. Its equation is .
For any angle measured from the positive -axis (anticlockwise), the point where the terminal side of the angle meets the unit circle has coordinates:
This means:
- The x-coordinate of the point equals
- The y-coordinate of the point equals
- The slope of the radius to that point equals
Because the radius is always 1, we don't need to divide by a hypotenuse — the coordinates are the trigonometric ratios.
Think of the unit circle like a clock face. As the "hand" sweeps around, the tip traces out a circle, and its horizontal and vertical positions at any moment are exactly and respectively.
Signs of Trig Functions in Each Quadrant
Because and are coordinates on the unit circle, their signs depend entirely on which quadrant the angle falls in.
| Quadrant | |||||
|---|---|---|---|---|---|
| I (0° to 90°) | + | + | + | + | + |
| II (90° to 180°) | − | + | − | + | − |
| III (180° to 270°) | − | − | − | − | + |
| IV (270° to 360°) | + | − | + | − | − |
Notice that , so it is positive whenever both have the same sign (Q1 and Q3) and negative when they differ (Q2 and Q4).
Use the mnemonic "All Students Take Calculus" to remember which functions are positive in each quadrant, going anticlockwise from Q1:
- All — all three functions positive (Q1)
- Sine — only sine positive (Q2)
- Tangent — only tangent positive (Q3)
- Cosine — only cosine positive (Q4)
Students often mix up Q2 and Q4. Remember: in Q2 you are still above the -axis so (the -value) is still positive, even though has turned negative.