🎡 The Unit Circle Compass: Allied Angle Formulas

The Story of the Magic Compass

Imagine you have a magic compass that always points somewhere on a circle. This circle has a radius of exactly 1 unit—we call it the Unit Circle.

Now, here’s the cool part: when your compass needle moves around this circle, it tells you the values of sine and cosine at any angle!

But there’s a secret shortcut. Instead of calculating every angle from scratch, mathematicians discovered allied angle formulas—magical rules that let you find trigonometric values for ANY angle if you just know what happens between 0° and 90°.

Think of it like this: If you know what’s in one corner of a square room, you can figure out what’s in ALL corners!

🧭 The Four Quadrants: Your Angle Neighborhoods

Before we dive in, let’s meet the four “neighborhoods” on our unit circle:

graph TD
    A[🏠 Quadrant I<br/>0° to 90°<br/>All positive] --> B[🏠 Quadrant II<br/>90° to 180°<br/>Only sin positive]
    B --> C[🏠 Quadrant III<br/>180° to 270°<br/>Only tan positive]
    C --> D[🏠 Quadrant IV<br/>270° to 360°<br/>Only cos positive]
    D --> A

Memory trick: “All Students Take Coffee”

• All (Quadrant I) - all ratios positive
• Students (Quadrant II) - sin positive
• Take (Quadrant III) - tan positive
• Coffee (Quadrant IV) - cos positive

📐 The Mirror Rule: Understanding Allied Angles

Allied angles are like mirrors. When you flip an angle across the x-axis, y-axis, or both, the trig ratios follow simple patterns.

The Big Secret:

• Angles involving 90° or 270° → sin and cos swap with each other
• Angles involving 180° or 360° → sin and cos stay as themselves

🔄 T-Ratios of Negative Theta (−θ)

The Story: Imagine walking backwards on our circle. If θ takes you counterclockwise, then −θ takes you clockwise by the same amount!

What happens:

• sin(−θ) = −sin θ (flips sign)
• cos(−θ) = cos θ (stays same)
• tan(−θ) = −tan θ (flips sign)

Why? When you go clockwise, your y-coordinate (sine) flips upside down, but your x-coordinate (cosine) stays the same!

Example

If sin 30° = 1/2 and cos 30° = √3/2, then:

• sin(−30°) = −1/2
• cos(−30°) = √3/2
• tan(−30°) = −1/√3

Memory Trick: “Cosine doesn’t care about direction!”

🌅 T-Ratios of (90° − θ): The Sunrise Angle

The Story: Imagine the sun rises at angle θ from the horizon. The angle from straight up (90°) down to the sun is (90° − θ). These two angles are complementary—they complete each other like puzzle pieces!

What happens:

• sin(90° − θ) = cos θ (they swap!)
• cos(90° − θ) = sin θ (they swap!)
• tan(90° − θ) = cot θ (they swap!)

Why? At 90° minus your angle, the roles of horizontal and vertical flip!

Example

If sin 60° = √3/2 and cos 60° = 1/2, then:

• sin(90° − 60°) = sin 30° = cos 60° = 1/2 ✓
• cos(90° − 60°) = cos 30° = sin 60° = √3/2 ✓

Memory Trick: “Complementary angles play swap-the-ratio!”

🌄 T-Ratios of (90° + θ): Just Past Sunrise

The Story: Now imagine going just a bit past 90°. You’ve entered Quadrant II, where only sine stays positive!

What happens:

• sin(90° + θ) = cos θ (swap, stays positive)
• cos(90° + θ) = −sin θ (swap, becomes negative)
• tan(90° + θ) = −cot θ (swap, becomes negative)

Why? You crossed into Quadrant II! Sine is happy there, but cosine becomes grumpy (negative).

Example

If sin 30° = 1/2 and cos 30° = √3/2, then:

• sin(90° + 30°) = sin 120° = cos 30° = √3/2
• cos(90° + 30°) = cos 120° = −sin 30° = −1/2

Memory Trick: “Past 90°, functions swap and cos goes negative!”

🌆 T-Ratios of (180° − θ): The Sunset Mirror

The Story: 180° is like looking straight left on our compass. When you subtract θ, you’re in Quadrant II, reflected across the y-axis!

What happens:

• sin(180° − θ) = sin θ (same sign!)
• cos(180° − θ) = −cos θ (flips sign!)
• tan(180° − θ) = −tan θ (flips sign!)

Why? The y-coordinate (sine) stays the same when you mirror across the y-axis, but x-coordinate (cosine) flips!

Example

If sin 45° = √2/2 and cos 45° = √2/2, then:

• sin(180° − 45°) = sin 135° = sin 45° = √2/2
• cos(180° − 45°) = cos 135° = −cos 45° = −√2/2

Memory Trick: “Mirror across y-axis: sine survives, cosine flips!”

🌑 T-Ratios of (180° + θ): Into the Shadow

The Story: 180° + θ takes you into Quadrant III—the shadow zone where both coordinates are negative! Only tangent is positive here.

What happens:

• sin(180° + θ) = −sin θ (negative!)
• cos(180° + θ) = −cos θ (negative!)
• tan(180° + θ) = tan θ (same!)

Why? In Quadrant III, both x and y are negative. But tan = sin/cos, so the negatives cancel out!

Example

If sin 60° = √3/2 and cos 60° = 1/2, then:

• sin(180° + 60°) = sin 240° = −√3/2
• cos(180° + 60°) = cos 240° = −1/2
• tan(180° + 60°) = tan 240° = tan 60° = √3

Memory Trick: “In the shadow (Q3), both sin and cos go dark, but tan stays bright!”

🌙 T-Ratios of (270° − θ): Approaching Midnight

The Story: 270° is straight down on our compass. Just before reaching it, you’re in Quadrant III.

What happens:

• sin(270° − θ) = −cos θ (swap and negative!)
• cos(270° − θ) = −sin θ (swap and negative!)
• tan(270° − θ) = cot θ (swap, stays positive!)

Why? Near 270°, functions swap (like near 90°), and since we’re in Q3, both base values go negative.

Example

If sin 30° = 1/2 and cos 30° = √3/2, then:

• sin(270° − 30°) = sin 240° = −cos 30° = −√3/2
• cos(270° − 30°) = cos 240° = −sin 30° = −1/2

Memory Trick: “270° zone = swap AND shadow!”

🌃 T-Ratios of (270° + θ): Just Past Midnight

The Story: Just past 270°, you enter Quadrant IV—where cosine wakes up positive again!

What happens:

• sin(270° + θ) = −cos θ (swap, still negative)
• cos(270° + θ) = sin θ (swap, becomes positive!)
• tan(270° + θ) = −cot θ (swap, negative)

Why? Functions swap near 270°. In Q4, x is positive (cos happy) but y is negative (sin sad).

Example

If sin 45° = √2/2 and cos 45° = √2/2, then:

• sin(270° + 45°) = sin 315° = −cos 45° = −√2/2
• cos(270° + 45°) = cos 315° = sin 45° = √2/2

Memory Trick: “Past 270°, swap happens; cos wakes up, sin stays asleep!”

🎯 The Master Pattern

Here’s the beautiful pattern that ties everything together:

The Golden Rules:

1. 90° and 270° → Functions SWAP (sin↔cos)
2. 180° and 360° → Functions STAY
3. Sign depends on which quadrant you land in!

🚀 Quick Practice Check

Question: What is cos(150°)?

Solution:

• 150° = 180° − 30°
• cos(180° − θ) = −cos θ
• cos(150°) = −cos 30° = −√3/2 ✓

Question: What is sin(300°)?

Solution:

• 300° = 270° + 30°
• sin(270° + θ) = −cos θ
• sin(300°) = −cos 30° = −√3/2 ✓

🎉 You Did It!

You’ve just learned the secret shortcuts of trigonometry! Instead of memorizing hundreds of values, you now know how to find ANY angle’s trig ratios using just the first quadrant.

Remember: The unit circle is your compass, allied angles are your shortcuts, and the quadrant tells you the sign!

Next time you see a scary angle like 315° or 225°, just smile and break it down. You’ve got this! 🌟