🎯⭐ INTERACTIVE LESSON

The Unit Circle

Learn step-by-step with interactive practice!

The Unit Circle - Complete Interactive Lesson

Part 1: Counting Method

Welcome to the Interactive Unit Circle Lesson!

In this interactive lesson, you'll master the unit circle through a simple counting technique that makes memorizing those tricky values super easy!

Click "Next" when you're ready to begin learning the counting method.

Step 1: Count from 0 to 4

The first step is simple: we're going to count from 0 to 4, including both 0 and 4.

These five numbers will become the foundation for remembering all the key sine and cosine values!

Now it's your turn!

Fill in the boxes below by counting from 0 to 4:

Step 2: Put Each Number Under a Square Root

Great job! Now we're going to take those numbers (0, 1, 2, 3, 4) and put each one under a square root symbol.

So we get: $\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}$

This might seem strange, but trust the process - it's about to make perfect sense!

Step 3: Simplify the Perfect Squares

Now we simplify any perfect squares in our list: $\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}$

Practice: Simplify the Square Roots

Select the correct simplified form for each square root:

Step 4: Divide Everything by 2

Almost there! Now we take our simplified values $(0, 1, \sqrt{2}, \sqrt{3}, 2)$ and divide each by 2.

🎉 You Did It!

These are the SINE values for 0°, 30°, 45°, 60°, and 90°!

$\sin(0°) = 0$

$\sin(30°) = \frac{1}{2}$

🌟 Part 1 Complete! 🌟

Congratulations! You've mastered the counting method for memorizing unit circle values.

Ready to practice on your own? Try the independent practice mode to reinforce what you've learned!

Part 2: Angles & Tables

Now Let's Learn the Angles!

The first quadrant of the unit circle has five key angles:

0°, 30°, 45°, 60°, and 90°

These correspond to the values we just calculated!

Let's practice identifying these angles.

Practice: Fill in the First Quadrant Angles

Enter the five key angles in the first quadrant in order:

Understanding "Sin UP" 📈

Look at how sine values change as we go from 0° to 90°:

[SINE_TABLE]

Sin goes UP! ⬆️ The values increase from 0 to 1 as the angle increases.

Remember: "Sin UP" - Sine values go UP from 0° to 90°!

Understanding "COAST DOWN" 📉

Now look at how COsine values change (remember: COSINE → COAST):

[COSINE_TABLE]

Cosine goes DOWN! ⬇️ The values decrease from 1 to 0 as the angle increases.

Remember: "COAST DOWN" - COsine values go DOWN from 90° to 0°!

Even better: Cosine values are just the sine values in reverse order!

🧠 The Ultimate Memory Trick! 🧠

Follow these simple steps to memorize all the values:

Step 1: Count from 0 to 4

$0,$

Part 3: What Is the Unit Circle?

What IS the Unit Circle? 🔵

Now that you know how to calculate the values, let's understand what the unit circle actually represents!

[UNIT_CIRCLE]

The Unit Circle is:

A circle with a radius of exactly 1 unit
Centered at the origin (0, 0) on a coordinate plane
Used to define sine and cosine for all angles

The "unit" in "unit circle" means the radius = 1. This makes all the math beautifully simple!

Understanding Coordinates on the Unit Circle 📍

[UNIT_CIRCLE_ANIMATION]

Here's the KEY concept that connects everything:

Every point on the unit circle has coordinates (x, y) where:

x = cos(θ) (the cosine of the angle)

y = sin(θ) (the sine of the angle)

So when we write a point like $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ on the unit circle at 30°:

Part 4: Complete Unit Circle

Welcome to the Complete Unit Circle! 🎯

You've mastered the first quadrant. Now let's expand to all four quadrants!

The unit circle has perfect symmetry that makes memorizing all the values much easier than you might think. Once you understand the patterns, you can figure out any angle on the unit circle.

What You'll Learn:

📐 How to extend the first quadrant to all four quadrants
🔄 Using reference angles to find any coordinate
➕➖ Understanding signs in each quadrant
🎯 Mastering the complete unit circle with 16 key angles

Reference Angles: Your Secret Weapon 🎯

A reference angle is the acute angle (0° to 90°) that any angle makes with the x-axis.

The Beautiful Truth: Every angle on the unit circle uses the same coordinate values as its reference angle in the first quadrant—just with different signs!

Finding Reference Angles:

Quadrant I (0° to 90°): Reference angle = the angle itself
Quadrant II (90° to 180°): Reference angle = 180° - angle
Quadrant III (180° to 270°): Reference angle = angle - 180°
Quadrant IV (270° to 360°): Reference angle = 360° - angle

Examples:

150° is in QII → reference angle = 180° - 150° = 30°
225° is in QIII → reference angle = 225° - 180° = 45°
315° is in QIV → reference angle = 360° - 315° = 45°

Practice: Find the Reference Angle 🎯

Remember: A perfect square is a number whose square root is a whole number.

$\sqrt{0} = 0$ (because $0^2 = 0$ )
$\sqrt{1} = 1$ (because $1^2 = 1$ )
$\sqrt{2}$ stays as $\sqrt{2}$ (not a perfect square)
$\sqrt{3}$ stays as $\sqrt{3}$ (not a perfect square)
$\sqrt{4} = 2$ (because $2^2 = 4$ )

$\frac{0}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{2}{2}$

Which simplifies to:

$0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$

$\sin(45°) = \frac{\sqrt{2}}{2}$

$\sin(60°) = \frac{\sqrt{3}}{2}$

BONUS TIP: For cosine values, just reverse the order!

$\cos(30°) = \frac{\sqrt{3}}{2}$

$\cos(45°) = \frac{\sqrt{2}}{2}$

$\cos(60°) = \frac{1}{2}$

Step 2: Take the square root of each number

$\sqrt{0}, \quad \sqrt{1}, \quad \sqrt{2}, \quad \sqrt{3}, \quad \sqrt{4}$

Step 3: Simplify the square roots

$0, \quad 1, \quad \sqrt{2}, \quad \sqrt{3}, \quad 2$

Step 4: Divide everything by 2

$0, \quad \frac{1}{2}, \quad \frac{\sqrt{2}}{2}, \quad \frac{\sqrt{3}}{2}, \quad 1$

Step 5: Match with the angles

Now see how these values become sine and cosine:

Angle	sin (UP ⬆️)	cos (DOWN ⬇️)
0°	$0$	$1$
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
45°	$\frac{\sqrt{2}}{2}$
60°	$\frac{\sqrt{3}}{2}$
90°	$1$	$0$

Remember the pattern:

✅ For Sine: "Sin UP" - Values go up from $0$ to $1$ (reading down the table)

✅ For Cosine: "COAST DOWN" - Values go down from $1$ to $0$ (reading down the table)

That's it! You now know the entire first quadrant of the unit circle! 🎉

🌟 Lesson Complete! 🌟

Congratulations! You've mastered the essential skills for memorizing the unit circle:

✅ Key Takeaways:

1. The Counting Method

Count 0, 1, 2, 3, 4
Take square roots
Divide by 2

2. The Five Key Angles

$0°, \quad 30°, \quad 45°, \quad 60°, \quad 90°$

3. The Sin UP Pattern ⬆️

Sine values increase from $0$ to $1$ as angles go from $0°$ to $90°$

4. The COAST DOWN Pattern ⬇️

Cosine values decrease from $1$ to $0$ as angles go from $0°$ to $90°$

🎯 What's Next?

Ready to practice on your own? Try the independent practice mode to reinforce what you've learned and master these patterns!

The x-coordinate $\frac{\sqrt{3}}{2}$ is $\cos(30°)$
The y-coordinate $\frac{1}{2}$ is $\sin(30°)$

Think of it this way: The angle tells you where to go, and the coordinates tell you the sine and cosine values!

The Special Right Triangles 📐

The unit circle values come from two special right triangles that you may already know:

30-60-90 Triangle:

Angles: 30°, 60°, 90°
Side ratio: $1 : \sqrt{3} : 2$
When we scale to fit the unit circle (radius = 1), we divide by 2
This gives us: $\frac{1}{2} : \frac{\sqrt{3}}{2} : 1$

45-45-90 Triangle:

Angles: 45°, 45°, 90°
Side ratio: $1 : 1 : \sqrt{2}$
When we scale to fit the unit circle, we divide by $\sqrt{2}$

These triangles are WHY our counting method works! ✨

The First Quadrant (0° to 90°) 🎯

Now it's your turn! Let's see if you can fill in the first quadrant of the unit circle.

[UNIT_CIRCLE_GAME]

Fill in all the angles and coordinates to complete the game. Remember:

Angles: 0°, 30°, 45°, 60°, 90°
Key coordinates: (1, 0), (0, 1), and the three special triangle points
Tip: Use the patterns from 30-60-90 and 45-45-90 triangles!

Why Does This All Work? 🤔

When you draw a radius from the origin to any point on the unit circle:

The angle is measured from the positive x-axis (going counterclockwise)
The horizontal distance from the origin is the cosine (x-coordinate)
The vertical distance from the origin is the sine (y-coordinate)

Because the radius is always 1, these distances ARE the sine and cosine values!

The Pythagorean Identity:

Since we're on a circle with radius 1, we always have:

$\sin^2(\theta) + \cos^2(\theta) = 1$

This is just the Pythagorean theorem: $x^2 + y^2 = r^2$ , where $r = 1$ !

This is why the unit circle is so powerful - it connects geometry, trigonometry, and algebra all in one beautiful circle! 🎨

🌟 You Now Understand the Unit Circle! 🌟

Congratulations! You now know:

✅ What the unit circle is (a circle with radius 1) ✅ How coordinates relate to sine and cosine ✅ Why the special triangles (30-60-90 and 45-45-90) give us our values ✅ How to read any point on the unit circle ✅ The connection between geometry and trigonometry

Up Next: You'll learn how to extend this to all four quadrants and work with angles beyond 90°!

You've mastered the foundation of the unit circle! 🎉

Let's practice finding reference angles! You need to answer 5 questions correctly in a row to proceed.

For each angle given, determine its reference angle (the acute angle it makes with the x-axis).

Signs in Each Quadrant ➕➖

Here's the pattern you need to remember:

Quadrant I (0° to 90°): Both x and y are positive ✅ (+, +)

Example: 30° → $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

Quadrant II (90° to 180°): x is negative, y is positive ✅ (-, +)

Example: 150° → $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

Quadrant III (180° to 270°): Both x and y are negative ❌ (-, -)

Example: 225° → $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

Quadrant IV (270° to 360°): x is positive, y is negative ✅ (+ , -)

Example: 315° → $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$

Memory Trick: "All Students Take Calculus"

All (QI): All positive
Students (QII): Sine (y) positive
Take (QIII): Tangent positive (both negative, so tangent positive)
Calculus (QIV): Cosine (x) positive

Let's Work Through Some Examples 📝

Example 1: Find the coordinates for 120°

Identify the quadrant: 120° is between 90° and 180°, so it's in Quadrant II
Find the reference angle: 180° - 120° = 60°
Look up 60° in QI: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
Apply QII signs (-, +): $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ ✅

Example 2: Find the coordinates for 240°

Identify the quadrant: 240° is between 180° and 270°, so it's in Quadrant III
Find the reference angle: 240° - 180° = 60°
Look up 60° in QI: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

Example 3: Find the coordinates for 330°

Identify the quadrant: 330° is between 270° and 360°, so it's in Quadrant IV
Find the reference angle: 360° - 330° = 30°
Look up 30° in QI: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

The Complete Unit Circle 🎯

Now it's your turn! Fill in the complete unit circle with all 16 key angles.

[FULL_UNIT_CIRCLE_GAME]

Fill in all the angles and coordinates to complete the game. Remember:

Use reference angles to determine the coordinate values
Apply the correct signs based on the quadrant
Quadrant I: Both positive (+, +)
Quadrant II: x negative, y positive (-, +)
Quadrant III: Both negative (-, -)
Quadrant IV: x positive, y negative (+, -)

The 16 Special Angles 🌟

Here's the complete set of angles you should know:

Quadrant I (0° to 90°):

0°, 30°, 45°, 60°, 90°

Quadrant II (90° to 180°):

120°, 135°, 150°, 180°

Quadrant III (180° to 270°):

210°, 225°, 240°, 270°

Quadrant IV (270° to 360°):

300°, 315°, 330°, 360° (same as 0°)

Pro Tip: Notice the symmetry! The angles in each quadrant mirror each other:

30° and 150° are both 30° from their nearest axis
45° and 135° and 225° and 315° are all 45° from their nearest axis
60° and 120° are both 30° from their nearest axis

Why Does the Symmetry Work? 🤔

The unit circle's symmetry comes from the definition of sine and cosine:

For any angle θ:

cos(θ) = x-coordinate = horizontal distance from origin
sin(θ) = y-coordinate = vertical distance from origin

The symmetry patterns:

Reflection across the y-axis (QI ↔ QII):
- x-coordinates become negative
- y-coordinates stay the same
- Example: cos(30°) = √3/2, cos(150°) = -√3/2
Reflection across the x-axis (QI ↔ QIV):
- x-coordinates stay the same
- y-coordinates become negative
- Example: sin(30°) = 1/2, sin(330°) = -1/2
Rotation by 180° (QI ↔ QIII):
- Both coordinates become negative
- Example: (√2/2, √2/2) at 45° becomes (-√2/2, -√2/2) at 225°

This is why you only need to memorize the first quadrant! 🎯

🎉 Congratulations! You've Mastered the Complete Unit Circle! 🎉

You now understand:

✅ All 16 key angles and their coordinates
✅ How to use reference angles
✅ The sign patterns in each quadrant
✅ The beautiful symmetry of the unit circle

What's Next?

Practice converting between degrees and radians
Learn how to find exact values for any trigonometric function
Apply the unit circle to solve trigonometric equations
Use the unit circle in calculus for derivatives and integrals

Keep practicing, and the unit circle will become second nature! 🌟

This gives us:

\frac{1}{\sqrt{2}} : \frac{1}{\sqrt{2}} : 1

(which simplifies to

\frac{\sqrt{2}}{2} : \frac{\sqrt{2}}{2} : 1

)

Apply QIII signs (-, -):

\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

✅

Apply QIV signs (+, -):

\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

✅