Definition: The unit circle is the circle of radius 1 centered at the origin; the point at angle t has coordinates (cos t, sin t), linking angles to trig values.
What is the unit circle?
It is the set of points (x, y) with x2 + y2 = 1. Start at (1, 0) and travel counterclockwise through an angle t. You land at the point (cos t, sin t).
Which values should you know?
- At 30 degrees: (√32, 12)
- At 45 degrees: (√22, √22)
- At 60 degrees: (12, √32)
Other angles use symmetry: reflect these across the axes and change signs by quadrant.
Worked example: for 210 degrees, the reference angle is 30 degrees in quadrant III, where both coordinates are negative. So cos 210 = -√3/2 and sin 210 = -1/2.
What is the usual mistake?
Getting the quadrant signs wrong. Mathbot’s memory aid: All, Sine, Tangent, Cosine are positive in quadrants I through IV.
Example: At 210 degrees the point is (-√3/2, -1/2)