🌐 Trig Unit Circle Study Guide
1. The Unit Circle Basics

Unit circle: circle of radius 1 centered at the origin.

Coordinates of a point on the circle: (cos θ, sin θ).

So:
o
cos θ = x-coordinate
o
sin θ = y-coordinate
2. Quadrantal Angles (the “easy” ones)
Angle (radians) Degrees cos θ sin θ
0 or 2π
0°/360° 1
0
π/2
90°
0
1
π
180°
-1
0
3π/2
270°
0
-1
✅ Easy to memorize: they’re just the axes.
3. Special Triangles to Memorize
45° (π/4)

Triangle: isosceles right (sides 1, 1, √2).

Coordinates: (√2/2, √2/2).

→ cos(π/4) = √2/2, sin(π/4) = √2/2.
30° (π/6) and 60° (π/3)

Triangle: 30–60–90 (sides 1, √3, 2).

π/6 (30°): (√3/2, 1/2).
→ cos(π/6) = √3/2, sin(π/6) = 1/2.

π/3 (60°): (1/2, √2/2).
→ cos(π/3) = 1/2, sin(π/3) = √3/2.
4. Symmetry & Quadrants

Quadrant I: all positive (0 to π/2).

Quadrant II: cos negative, sin positive.

Quadrant III: both negative.

Quadrant IV: cos positive, sin negative.
👉 Rule: The absolute values of sine/cosine are the same as the reference angle
(distance from x-axis). The sign depends on quadrant.
5. Handy Mnemonics


"All Students Take Calculus" (Quadrants I–IV):
o
I: All (sin, cos, tan positive)
o
II: Students (sin positive)
o
III: Take (tan positive)
o
IV: Calculus (cos positive)
1, 2, √3 over 2, 1 pattern:
o
For sin: go up as angle increases (0, 30, 45, 60, 90 → 0, 1/2, √2/2, √3/2,
1).
o
For cos: go down in reverse order (1, √3/2, √2/2, 1/2, 0).
6. Summary Table (First Quadrant, then reflect)
Degrees Radians cos θ sin θ
0°
0
1
30°
π/6
√3/2 1/2
45°
π/4
√2/2 √2/2
60°
π/3
1/2
0
√3/2
Degrees Radians cos θ sin θ
90°
π/2
0
1
👉 Use symmetry for the rest (120°, 135°, etc.).
7. Quick Checks

cos²θ + sin²θ = 1 (Pythagorean identity).

Sin = y, Cos = x, Tan = y/x.

tan(θ) = sin(θ)/cos(θ).