The Unit Circle: Special Angles You Must Memorize
The unit circle is the source of all trigonometry. Master these five special angles and signs by quadrant — everything else builds from here.
The unit circle is non-negotiable
On the unit circle (radius 1), the point at angle θ has coordinates (cos θ, sin θ). The Pre-Calc CBE expects you to recall the special-angle values WITHOUT a calculator — for sin, cos, and tan at 0, π/6, π/4, π/3, π/2.
The five must-know values
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 (30°) | 1/2 | √3/2 | 1/√3 |
| π/4 (45°) | √2/2 | √2/2 | 1 |
| π/3 (60°) | √3/2 | 1/2 | √3 |
| π/2 (90°) | 1 | 0 | undefined |
Signs by quadrant: ASTC
"All Students Take Calculus":
- Q1: ALL positive
- Q2: only SINE positive
- Q3: only TANGENT positive
- Q4: only COSINE positive
Reference angle
To find trig of angles outside Q1: find the reference angle (acute angle to the nearest x-axis), look up its value, apply the quadrant sign.
Example: cos(5π/6) — Q2, ref angle π/6, cos(π/6) = √3/2, in Q2 cos is negative → cos(5π/6) = −√3/2.
Check yourself
📌 Unit circle recall
What is sin(7π/6)?
7π/6 is in Q3 (sin negative); reference angle π/6 → sin = 1/2; result = −1/2.
Practice with CBE-style practice questions
Drill the unit circle in Pre-Calc Sem B practice.