Question

How can you find the measures of the angles (in degrees and radians) by thinking about fraction
pieces of the Unit Circle?
What are some patterns you notice in the coordinates of the points?
Are there any other patterns you notice that would help you remember all this information on
the Unit Circle?

Answer 1

Think of the unit circle as a circle of radius 1 centered at the origin. One full turn around the circle is 360° = 2π radians. If you divide the circle into equal fraction pieces, each piece is that fraction of 360° or 2π.

- If you divide into 12 pieces: each piece = 360/12 = 30° = π/6.
- Into 8 pieces: each = 45° = π/4.
- Into 6 pieces: each = 60° = π/3.
So angles you commonly memorize are 0, 30°, 45°, 60°, 90° (i.e. 0, π/6, π/4, π/3, π/2) and their multiples up to 360° (2π).

How to get coordinates (cos θ, sin θ)
- Every point on the unit circle at angle θ has coordinates (cos θ, sin θ).
- For the special angles in the first quadrant you can get coordinates from two special right triangles:
- 45° (45–45–90): normalized legs give cos 45° = sin 45° = √2/2.
- 30° and 60° (30–60–90): normalized give cos 30° = √3/2, sin 30° = 1/2; cos 60° = 1/2, sin 60° = √3/2.
- First-quadrant list (degrees, radians, coordinates):
- 0° = 0 = (1, 0)
- 30° = π/6 = (√3/2, 1/2)
- 45° = π/4 = (√2/2, √2/2)
- 60° = π/3 = (1/2, √3/2)
- 90° = π/2 = (0, 1)

Patterns in the coordinates
- Absolute values of common coordinates come from the set {0, 1/2, √2/2, √3/2, 1}. Every special-angle coordinate is a pair chosen from that set.
- The sin values for multiples of 30° follow a symmetric pattern:
0, 1/2, √3/2, 1, √3/2, 1/2, 0 (for 0°,30°,60°,90°,120°,150°,180°)
- The cos values are the sin values shifted by 90° (cos θ = sin(90° − θ)), so you only need to memorize one sequence.
- Symmetry (use reference angles):
- Quadrant I: ( + , + )
- Quadrant II: ( − , + )
- Quadrant III: ( − , − )
- Quadrant IV: ( + , − )
Use sign rules and the reference angle (the acute angle the terminal ray makes with the x-axis) to get coordinates in any quadrant.
- Even/odd properties:
- cos(−θ) = cos θ (even), sin(−θ) = −sin θ (odd).
- cos(π − θ) = −cos θ, sin(π − θ) = sin θ, etc.

Memory aids / tricks
- Memorize only the first-quadrant coordinates for 0°, 30°, 45°, 60°, 90°. Then get every other special angle by symmetry and sign changes.
- “All Students Take Calculus” (or “CAST”) to remember which trig functions are positive in which quadrant.
- Think in fractions: e.g. 30° = 1/12 of the circle (π/6 = 1/6 of π), 45° = 1/8 of the circle (π/4 = 1/4 of π). This gives quick degree ↔ radian conversions: multiply fraction × 360° or × 2π.
- A quick numeric checklist for sine at 30° increments: 0, 1/2, √3/2, 1, √3/2, 1/2, 0 — reflect and change signs for other quadrants.

Examples
- 150° = 180° − 30° → reference angle 30°, quadrant II so cos negative, sin positive → (−√3/2, 1/2).
- 330° = 360° − 30° → reference 30°, quadrant IV → (√3/2, −1/2).
- π/4 + π/2 = 3π/4 → (−√2/2, √2/2).

Summary
- Divide the circle into equal fractional pieces to get angle measures in degrees and radians.
- Memorize the first-quadrant coordinates (0°, 30°, 45°, 60°, 90°) from the 30–60–90 and 45–45–90 triangles.
- Use symmetry, sign rules, and reference angles to get coordinates for all other angles.