Unit Circle Study Guide

🟢 #1 UNIT CIRCLE (Labeling + Coordinates + Radians)

📌 What the Unit Circle Is

  • A circle with radius = 1

  • Center at (0, 0)

  • Every point on it is:
     (cos θ, sin θ)

📍 Key Angles (YOU MUST MEMORIZE)

First Quadrant (the “base” angles)

Degrees

Radians

Coordinates (cos, sin)

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)

🔁 How to Label the Whole Circle (SUPER IMPORTANT)

Use symmetry + signs:

Quadrants (ASTC Rule)

  • Q1: All positive

  • Q2: Sin positive

  • Q3: Tan positive

  • Q4: Cos positive

👉 Mnemonic: “All Students Take Calculus”

💡 Trick for Coordinates

Only memorize Q1, then:

  • Q2: flip x (negative)

  • Q3: both negative

  • Q4: flip y (negative)

Example:

  • 45° → (√2/2, √2/2)

  • 135° → (-√2/2, √2/2)

  • 225° → (-√2/2, -√2/2)

  • 315° → (√2/2, -√2/2)

🧠 Radian Pattern Trick

Denominators follow pattern:

  • π/6, π/4, π/3, π/2

Numerators increase as you move around:
 π/6 → 2π/6 → 3π/6 → etc.

🟡 #2 Evaluating Trig WITHOUT Calculator

📌 Basics

  • sin = y

  • cos = x

  • tan = y/x

  • sec = 1/cos

  • csc = 1/sin

  • cot = x/y

🧠 Example 1:

cos(-5π/4)

Step 1: Make positive
 -5π/4 → same as 3π/4

Step 2: Find point
 3π/4 = (-√2/2, √2/2)

Step 3: cosine = x
 ✅ Answer: -√2/2

🧠 Example 2:

sec(-60°)

Step 1: cos(-60°) = cos(60°) = 1/2
 Step 2: sec = 1/cos
 → 1 ÷ (1/2) = 2

✅ Answer: 2

🔥 BIG TIPS

  • Negative angle = go clockwise

  • Even/Odd rules:

    • cos(-θ) = cos(θ)

    • sin(-θ) = -sin(θ)

  • Always convert to a known angle

🔵 #3 Converting Radians ↔ Degrees

📌 Formulas

Use:

180∘=π radians180^\circ = \pi \text{ radians}180∘=π radians

➡ Degrees → Radians

Multiply by π/180

Example:
 170°

= 170 × π/180
 = 17π/18

✅ Answer: 17π/18

⬅ Radians → Degrees

Multiply by 180/π

Example:
 7π/18

= 7π/18 × 180/π
 = 70°

✅ Answer: 70°

🧠 Fraction Reduction Tip

Cancel BEFORE multiplying!

Example:
 180/18 = 10

🟣 #4 Lowest Non-Negative Equivalent Angle

📌 Rule

Add 2π until positive

🧠 Example:

-2π/3

  • 2π = -2π/3 + 6π/3
     = 4π/3

✅ Answer: 4π/3

🔥 Shortcut

Think:
 “How far around the circle?”

🔴 #5 Open Response Strategies

🧠 Example 1:

Smallest positive radian with tan = -1

Step 1: tan = -1 happens at 45° reference
 Step 2: tangent negative in Q2 & Q4
 Step 3: smallest positive → Q2

45° → π/4
 Q2 → π - π/4 = 3π/4

✅ Answer: 3π/4

🧠 Example 2:

Smallest positive degree where cot is undefined

Step 1: cot = x/y
 Undefined when y = 0

That happens at:

  • 0°

  • 180°

Smallest positive →

✅ Answer: 0° is not positive → 180°

🚀 FINAL MEMORY HACKS

🔥 1. Square Root Pattern

Coordinates follow:

  • √3/2, √2/2, 1/2 (for cosine)

  • reversed for sine

🔥 2. Hand Trick (for sine)

Left hand = 0° → 90°
 Fold finger = gives √ pattern

🔥 3. Always Find:

  1. Reference angle

  2. Quadrant

  3. Sign

🔥 4. Reciprocals

  • sec = flip cos

  • csc = flip sin

  • cot = flip tan