🎯 Multiple Angle Formulas: The Magic Multiplier
Imagine you have a recipe for making a cake. Now, what if you wanted to make a DOUBLE cake, or a TRIPLE cake, or even just HALF a cake? You’d need to adjust your recipe! That’s exactly what multiple angle formulas do with angles in trigonometry.
🌟 The Big Idea
Think of angles like pizza slices. If you know what happens with ONE slice (sin θ, cos θ, tan θ), these formulas tell you what happens when you:
- Double the slice (2θ)
- Triple the slice (3θ)
- Cut it in half (θ/2)
- Add three slices together (A + B + C)
🎂 Double Angle for Sine
The Story
Picture a swing going back and forth. If you push it TWICE as far, what happens? It’s not just double the height — it’s more complicated! The swing follows a special pattern.
The Formula
sin(2θ) = 2 · sin(θ) · cos(θ)
What This Means
To find sine of a DOUBLE angle, you need BOTH sine AND cosine of the original angle, then multiply them together and double it.
Example
If sin(30°) = 1/2 and cos(30°) = √3/2, then:
sin(60°) = sin(2 × 30°)
= 2 × (1/2) × (√3/2)
= 2 × √3/4
= √3/2 ✓
Memory Trick 💡
“2 SC” — Two times Sine times Cosine!
🍪 Double Angle for Cosine
The Story
Imagine a seesaw. Cosine tells you how “balanced” it is. When you double the tilt, there are THREE different ways to describe the new balance!
The Three Formulas
cos(2θ) = cos²(θ) - sin²(θ) ← Version 1
cos(2θ) = 2cos²(θ) - 1 ← Version 2
cos(2θ) = 1 - 2sin²(θ) ← Version 3
Why Three Versions?
- Version 1: When you know both sin and cos
- Version 2: When you only know cos
- Version 3: When you only know sin
Example
Using cos(45°) = √2/2:
cos(90°) = cos(2 × 45°)
= 2cos²(45°) - 1
= 2 × (√2/2)² - 1
= 2 × (1/2) - 1
= 1 - 1
= 0 ✓
Memory Trick 💡
“C² minus S²” for the main formula!
🔄 Double Angle for Tangent
The Story
Tangent is like a ladder leaning against a wall. Double the angle? The ladder gets MUCH steeper, but in a special way!
The Formula
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Important Warning ⚠️
This formula BREAKS when tan²(θ) = 1 (that’s when θ = 45°). At that point, you’d divide by zero!
Example
If tan(30°) = 1/√3:
tan(60°) = tan(2 × 30°)
= 2 × (1/√3) / (1 - (1/√3)²)
= (2/√3) / (1 - 1/3)
= (2/√3) / (2/3)
= (2/√3) × (3/2)
= 3/√3
= √3 ✓
Memory Trick 💡
“2T over (1 minus T²)” — Think of it as “Two Tangents divided by One minus Tangent squared”
🎪 Triple Angle Formulas
The Story
If double was cool, TRIPLE is even more powerful! It’s like tripling your speed on a roller coaster — things get wild!
The Formulas
sin(3θ) = 3sin(θ) - 4sin³(θ)
cos(3θ) = 4cos³(θ) - 3cos(θ)
tan(3θ) = (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))
Pattern to Notice
- Sine: Starts with 3 × sin, subtracts 4 × sin³
- Cosine: Starts with 4 × cos³, subtracts 3 × cos
- See how 3 and 4 swap places? 🔄
Example
Using sin(30°) = 1/2:
sin(90°) = sin(3 × 30°)
= 3(1/2) - 4(1/2)³
= 3/2 - 4(1/8)
= 3/2 - 1/2
= 1 ✓
Memory Trick 💡
“3-4 for Sine, 4-3 for Cosine” — the coefficients flip!
✂️ Half Angle Formulas
The Story
Now let’s go the OTHER direction! What if you have a BIG angle and need the HALF? It’s like cutting a pizza slice in two — but mathematically!
The Formulas
sin(θ/2) = ±√[(1 - cos(θ))/2]
cos(θ/2) = ±√[(1 + cos(θ))/2]
tan(θ/2) = ±√[(1 - cos(θ))/(1 + cos(θ))]
The ± Sign Mystery
The ± tells you: “Check which quadrant θ/2 is in!”
- If θ/2 is in Quadrant 1 or 2: sin is positive
- If θ/2 is in Quadrant 1 or 4: cos is positive
Example
Find sin(15°) using half of 30°:
sin(15°) = sin(30°/2)
= √[(1 - cos(30°))/2]
= √[(1 - √3/2)/2]
= √[(2 - √3)/4]
= (√(2 - √3))/2
≈ 0.259 ✓
Alternative Tan Formulas
tan(θ/2) = sin(θ)/(1 + cos(θ))
tan(θ/2) = (1 - cos(θ))/sin(θ)
These avoid the ± problem!
🔬 Sub-Multiple Angle Formulas
The Story
These are like the “reverse engineering” formulas. Given sin(θ), cos(θ), or tan(θ), we find the values at θ/3, θ/4, or any fraction!
The Key Insight
Sub-multiple formulas are extensions of half-angle formulas. For example:
For θ/3 (one-third angle):
sin(θ) = 3sin(θ/3) - 4sin³(θ/3)
cos(θ) = 4cos³(θ/3) - 3cos(θ/3)
Example
To find sin(10°), use sin(30°) = 1/2:
Let x = sin(10°)
Then: 1/2 = 3x - 4x³
This gives: 8x³ - 6x + 1 = 0
Solving: x = sin(10°) ≈ 0.1736
When to Use
- Finding exact values of unusual angles (like 10°, 15°, 18°)
- Simplifying complex expressions
- Solving trigonometric equations
🎭 Compound Angle (Three Angles)
The Story
What if you need to add THREE angles together? It’s like combining three different dance moves into one smooth motion!
The Formula for sin(A + B + C)
sin(A + B + C) = sin(A)cos(B)cos(C)
+ cos(A)sin(B)cos(C)
+ cos(A)cos(B)sin(C)
- sin(A)sin(B)sin(C)
The Formula for cos(A + B + C)
cos(A + B + C) = cos(A)cos(B)cos(C)
- cos(A)sin(B)sin(C)
- sin(A)cos(B)sin(C)
- sin(A)sin(B)cos(C)
The Formula for tan(A + B + C)
tan(A + B + C) = (tan A + tan B + tan C - tan A·tan B·tan C)
────────────────────────────────────────────
(1 - tan A·tan B - tan B·tan C - tan C·tan A)
Example
Verify: sin(30° + 45° + 15°) = sin(90°) = 1
Let A = 30°, B = 45°, C = 15°
Using the formula with known values:
sin(90°) = sin(30°)cos(45°)cos(15°)
+ cos(30°)sin(45°)cos(15°)
+ cos(30°)cos(45°)sin(15°)
- sin(30°)sin(45°)sin(15°)
= 1 ✓
Memory Trick 💡
For sine: “One sine at a time, minus all sines”
- Three terms have exactly ONE sine
- Last term has ALL sines (and subtract it)
For cosine: “All cosines minus pairs of sines”
- First term is all cosines
- Other terms pair up sines
🗺️ Visual Summary
graph TD A["Basic Angle θ"] --> B["Double 2θ"] A --> C["Triple 3θ"] A --> D["Half θ/2"] A --> E["Sub-Multiple θ/n"] B --> F["sin 2θ = 2 sin θ cos θ"] B --> G["cos 2θ = cos²θ - sin²θ"] B --> H["tan 2θ = 2tan θ/1-tan²θ"] C --> I["sin 3θ = 3sin θ - 4sin³θ"] C --> J["cos 3θ = 4cos³θ - 3cos θ"] D --> K["Uses √ formulas"] E --> L["Reverse of multiple"]
🎯 Quick Reference Table
| Formula Type | Sine | Cosine |
|---|---|---|
| Double (2θ) | 2 sin θ cos θ | cos²θ - sin²θ |
| Triple (3θ) | 3 sin θ - 4 sin³θ | 4 cos³θ - 3 cos θ |
| Half (θ/2) | ±√[(1-cos θ)/2] | ±√[(1+cos θ)/2] |
🏆 You’ve Got This!
These formulas might look scary at first, but remember:
- They’re all connected — half-angle and double-angle are inverses!
- Practice with the examples above
- The patterns (like 3-4, 4-3) make them easier to remember
You now have the tools to find the sine, cosine, and tangent of ANY angle, as long as you know a related angle! That’s powerful stuff! 🚀
“Mathematics is not about numbers, equations, or algorithms: it is about understanding.” — William Paul Thurston