The Unit Circle: Your Secret Map to Trigonometry 🎯

Imagine you have a magical pizza that’s exactly 1 unit wide. Cut it perfectly round, put it at the center of a piece of paper, and BOOM — you’ve got the Unit Circle! This simple circle is the key that unlocks all of trigonometry.

🍕 What is a Radian?

Think about wrapping a piece of string around your pizza. If you take a string as long as the radius (the distance from center to edge) and wrap it along the edge, the angle you create is called 1 radian.

graph TD
    A[🎯 Center of Circle] --> B[Edge of Circle]
    B --> C[Wrap radius along edge]
    C --> D[That angle = 1 RADIAN!]

The Big Discovery

• A full circle = 2π radians (about 6.28 radians)
• Half circle = π radians (about 3.14 radians)
• Quarter circle = π/2 radians (about 1.57 radians)

Why 2π? Because the distance around ANY circle is 2π times the radius! So if you wrap the radius around the whole edge, it fits 2π times.

🔄 Converting Degrees and Radians

Here’s the magic formula — it’s like translating between two languages!

Degrees → Radians

Multiply by π/180

Radians → Degrees

Multiply by 180/π

Example: π/4 radians × (180/π) = 45°

Quick Memory Trick 🧠

• π = 180° (memorize this ONE thing!)
• Need 90°? That’s half of 180°, so it’s π/2
• Need 60°? That’s a third of 180°, so it’s π/3

🤔 Why Radians Are Useful

You might wonder: “Why learn a new way to measure angles when degrees work fine?”

The Short Answer

Radians make math formulas WAY simpler!

Real-World Example

When you spin a wheel:

• In degrees: Distance = (angle × π × radius) / 180 😵
• In radians: Distance = angle × radius 😊

See? No messy fractions! Radians are the “natural” language of circles.

Where Radians Shine ✨

1. Calculus — derivatives of sin and cos only work cleanly in radians
2. Physics — angular velocity, rotation, waves
3. Computer graphics — all programming uses radians
4. Engineering — motors, gears, oscillations

⭕ The Unit Circle Definition

The Unit Circle is simply a circle with:

• Center at the origin (0, 0)
• Radius of exactly 1 unit

graph TD
    A[Unit Circle] --> B[Center at 0,0]
    A --> C[Radius = 1]
    A --> D[Every point is 1 unit from center]

Why Radius = 1?

When the radius is 1, something magical happens: the coordinates of any point on the circle directly give us sine and cosine values! No extra calculations needed.

📍 Unit Circle Coordinates

Here’s where it gets beautiful! Every point on the unit circle can be written as (cos θ, sin θ) where θ is the angle from the positive x-axis.

The Famous Coordinates

Pattern to Remember 🎯

For the first quadrant special angles (30°, 45°, 60°):

The coordinates use these values: 0, 1/2, √2/2, √3/2, 1

Going counterclockwise from 0°:

• Cosine (x): goes DOWN → 1, √3/2, √2/2, 1/2, 0
• Sine (y): goes UP → 0, 1/2, √2/2, √3/2, 1

📊 Sine and Cosine on the Unit Circle

Here’s the core secret of the unit circle:

For ANY angle θ:

• cos θ = the x-coordinate of the point
• sin θ = the y-coordinate of the point

graph TD
    A[Point on Unit Circle] --> B[x-coordinate = cos θ]
    A --> C[y-coordinate = sin θ]
    B --> D[How far LEFT or RIGHT]
    C --> E[How far UP or DOWN]

Visual Story 🎬

Imagine an ant walking counterclockwise around the unit circle starting from (1, 0):

1. At 0°: Ant is at (1, 0) → cos 0° = 1, sin 0° = 0
2. At 90°: Ant climbs to (0, 1) → cos 90° = 0, sin 90° = 1
3. At 180°: Ant reaches (-1, 0) → cos 180° = -1, sin 180° = 0
4. At 270°: Ant drops to (0, -1) → cos 270° = 0, sin 270° = -1

Why This Works

Remember the Pythagorean theorem? For any point (x, y) on the unit circle:

x² + y² = 1 (radius squared)

Since x = cos θ and y = sin θ:

cos²θ + sin²θ = 1 ← The most important trig identity!

📐 Tangent on the Unit Circle

Tangent is simply the ratio of sine to cosine:

tan θ = sin θ / cos θ = y / x

Visual Meaning

Tangent tells you the slope of the line from the origin to your point on the circle!

When Tangent Breaks 💥

• At 90° and 270°, cos θ = 0
• You can’t divide by zero!
• So tan 90° and tan 270° are undefined
• The tangent line shoots off to infinity

🧭 Signs by Quadrant: ASTC

Here’s a life-saving memory trick! ASTC tells you which functions are positive in each quadrant:

         90°
          |
    S     |     A
  (sin+)  |  (ALL+)
          |
180° -----+------ 0°
          |
    T     |     C
  (tan+)  |  (cos+)
          |
         270°

ASTC Memory Tricks

• All Students Take Calculus
• Add Sugar To Coffee
• All Silly Tom Cats

The Logic Behind It

Example

What’s the sign of sin 150°?

1. 150° is in Quadrant II
2. Quadrant II is “S” — only Sine is positive
3. So sin 150° is POSITIVE ✓

🪞 Symmetry in the Unit Circle

The unit circle has beautiful symmetry that makes calculations easier!

Four Types of Symmetry

1. X-axis Symmetry (Reflection over horizontal)

If (x, y) is on the circle, so is (x, -y)

• sin(-θ) = -sin θ (y flips sign)
• cos(-θ) = cos θ (x stays same)

Example: sin(-30°) = -sin(30°) = -1/2

2. Y-axis Symmetry (Reflection over vertical)

If (x, y) is on the circle, so is (-x, y)

• sin(180° - θ) = sin θ
• cos(180° - θ) = -cos θ

Example: sin(150°) = sin(180° - 30°) = sin(30°) = 1/2

3. Origin Symmetry (Reflection through center)

If (x, y) is on the circle, so is (-x, -y)

• sin(180° + θ) = -sin θ
• cos(180° + θ) = -cos θ

Example: cos(210°) = cos(180° + 30°) = -cos(30°) = -√3/2

4. Reference Angles

Any angle can be reduced to a reference angle (0° to 90°) using symmetry!

graph TD
    A[Any Angle θ] --> B[Find Reference Angle]
    B --> C[Use ASTC for Sign]
    C --> D[Get Exact Value!]

Symmetry Cheat Sheet

🎓 Putting It All Together

The unit circle is your complete trigonometry toolkit:

1. Radians measure angles using the radius as your ruler
2. Convert easily: multiply by π/180 or 180/π
3. Unit circle has radius 1, centered at origin
4. Coordinates (cos θ, sin θ) give you trig values instantly
5. Sine = y-coordinate (up/down)
6. Cosine = x-coordinate (left/right)
7. Tangent = y/x = slope
8. ASTC tells you positive functions by quadrant
9. Symmetry lets you find any angle from reference angles

The Power Move 💪

Once you memorize just 5 points in the first quadrant (0°, 30°, 45°, 60°, 90°), you can find the sine, cosine, and tangent of ANY angle using ASTC and symmetry!

🌟 You’ve Got This!

The unit circle might look intimidating at first, but it’s really just:

• A circle with radius 1
• Where x = cosine and y = sine
• With beautiful symmetry that makes math easier

Master the first quadrant, understand ASTC, and you’ll have trigonometry at your fingertips! 🚀