General Angle Formulas

The Magic Formulas: General Angle Formulas in Trigonometry

The Big Picture: Building Blocks Like LEGO

Imagine you have LEGO blocks. With just a few basic pieces, you can build amazing things! Trigonometry works the same way. With simple sin and cos values, we can build formulas for ANY angle!

Today’s adventure: Learning the secret recipes that let us find sin and cos for angles like 15°, 18°, 36°, and 75° - angles that seem impossible at first!

1. Sin of nθ Formula: The Multiplication Pattern

The Story

Think of a photocopier. You put in one piece of paper (your angle θ), and it makes copies. But these copies stack in a special way!

The Formula

When you want sin(nθ) - that’s sin of an angle multiplied by n - there’s a pattern:

For sin(2θ):

sin(2θ) = 2 · sin(θ) · cos(θ)

For sin(3θ):

sin(3θ) = 3·sin(θ) - 4·sin³(θ)

For sin(4θ):

sin(4θ) = 4·cos³(θ)·sin(θ) - 4·cos(θ)·sin³(θ)

Simple Example

If θ = 30°, then sin(30°) = 1/2 and cos(30°) = √3/2

Finding sin(60°) using the formula:

sin(2×30°) = 2 × (1/2) × (√3/2)
           = 2 × √3/4
           = √3/2 ✓

This matches what we know: sin(60°) = √3/2!

The General Pattern

sin(nθ) = Σ combinations of sin and cos powers

The formula uses something called binomial coefficients - fancy words for counting patterns!

2. Cos of nθ Formula: The Partner Pattern

The Story

Cos is sin’s dance partner. They move together! Where sin goes up, cos goes sideways. Their formulas look similar but have their own steps.

The Formula

For cos(2θ):

cos(2θ) = cos²(θ) - sin²(θ)
        = 2cos²(θ) - 1
        = 1 - 2sin²(θ)

(Three ways to write the same thing!)

For cos(3θ):

cos(3θ) = 4·cos³(θ) - 3·cos(θ)

For cos(4θ):

cos(4θ) = 8·cos⁴(θ) - 8·cos²(θ) + 1

Simple Example

If θ = 30°, cos(30°) = √3/2

Finding cos(60°):

cos(2×30°) = 2×cos²(30°) - 1
           = 2×(√3/2)² - 1
           = 2×(3/4) - 1
           = 3/2 - 1
           = 1/2 ✓

Yes! cos(60°) = 1/2, and our formula worked!

3. Chebyshev Polynomials: The Special Helpers

The Story

Meet the Chebyshev family - a group of polynomial helpers named T₀, T₁, T₂, T₃… They have a superpower: cos(nθ) = Tₙ(cos θ)

They’re like translators between the world of angles and the world of algebra!

The Magic Connection

T₀(x) = 1
T₁(x) = x
T₂(x) = 2x² - 1
T₃(x) = 4x³ - 3x
T₄(x) = 8x⁴ - 8x² + 1

The Building Rule

Each Chebyshev polynomial is built from the two before it:

Tₙ(x) = 2x·Tₙ₋₁(x) - Tₙ₋₂(x)

Simple Example

Let’s check T₃(x) = 4x³ - 3x when x = cos(30°) = √3/2

T₃(√3/2) = 4×(√3/2)³ - 3×(√3/2)
         = 4×(3√3/8) - 3√3/2
         = 3√3/2 - 3√3/2
         = 0

And cos(3×30°) = cos(90°) = 0 ✓

The Chebyshev polynomial gave us the right answer!

Why They Matter

• Fast calculations for computers
• Pattern recognition in cos formulas
• Used in approximation and signal processing

4. Values of 15° and 75°: The Half-Step Angles

The Story

15° and 75° are like the “in-between” friends. 15° is halfway between 0° and 30°. And 15° + 75° = 90° (they’re complementary buddies!)

Finding sin(15°) and cos(15°)

Method: Use 45° - 30° = 15°

Using the difference formula:

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

sin(15°) = sin(45° - 30°)

= sin(45°)cos(30°) - cos(45°)sin(30°)
= (√2/2)(√3/2) - (√2/2)(1/2)
= √6/4 - √2/4
= (√6 - √2)/4

cos(15°) = cos(45° - 30°)

= cos(45°)cos(30°) + sin(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2)/4

The 75° Shortcut

Since 75° + 15° = 90°:

sin(75°) = cos(15°) = (√6 + √2)/4
cos(75°) = sin(15°) = (√6 - √2)/4

Easy Memory Trick

• 15°: The SMALLER angle gets the MINUS sign: (√6 - √2)/4 for sin
• 75°: The BIGGER angle gets the PLUS sign: (√6 + √2)/4 for sin

5. Values of 18° and 36°: The Golden Ratio Angles

The Story

These are the STAR angles - literally! They appear in the points of a regular pentagon (5-pointed star). And they have a magical secret ingredient: the Golden Ratio φ = (1 + √5)/2 ≈ 1.618

Finding cos(36°)

Start with the fact that 5×36° = 180°

So: sin(2×36°) = sin(180° - 3×36°) = sin(3×36°)

Using our earlier formulas and solving:

cos(36°) = (1 + √5)/4 = φ/2

Where φ = golden ratio!

Finding sin(36°)

Using sin²(36°) + cos²(36°) = 1:

sin(36°) = √(10 - 2√5)/4

The 18° Connection

Since 36° = 2×18°, we use half-angle formulas:

cos(18°) = √(10 + 2√5)/4
sin(18°) = (√5 - 1)/4

Fun fact: sin(18°) = (√5 - 1)/4 = 1/(2φ)

The Pattern Chart

Notice: 18° and 72° are swapped, and 36° and 54° are swapped! (They’re complementary pairs!)

The Flow of Everything

graph TD
    A["Basic Angles<br/>30°, 45°, 60°"] --> B["Multiple Angle<br/>Formulas"]
    B --> C["sin nθ formula"]
    B --> D["cos nθ formula"]
    D --> E["Chebyshev<br/>Polynomials"]
    A --> F["Sum/Difference<br/>Formulas"]
    F --> G["15° and 75°<br/>values"]
    A --> H["Pentagon<br/>Geometry"]
    H --> I["18° and 36°<br/>values"]
    I --> J["Golden Ratio φ"]

Summary: Your New Superpowers

1. sin(nθ) - Expand using powers of sin and cos
2. cos(nθ) - Same idea, different pattern
3. Chebyshev - cos(nθ) = Tₙ(cos θ), with recursive building
4. 15° & 75° - Use (√6 ± √2)/4, from 45° - 30°
5. 18° & 36° - Involve the Golden Ratio, from pentagon geometry

You now have the tools to find trig values for angles that seemed impossible! These aren’t just math tricks - they’re used in:

• Computer graphics (rotation)
• Music (sound waves)
• Engineering (bridges, buildings)
• Nature (the golden ratio is everywhere!)

Keep practicing, and these formulas will become as natural as counting! 🌟