Basic Equations and R-Formula

🎯 Trigonometric Equations: Your Treasure Map to Finding Angles!

The Big Picture: What’s This All About?

Imagine you’re a detective 🔍. Someone tells you that when they walked a certain angle, they ended up at a specific spot. Your job? Find that angle!

That’s exactly what trigonometric equations are about. You know the result (like sin θ = 0.5), and you need to find the angle θ that makes it true.

Think of it like this:

• Regular equation: 2x = 6 → What number times 2 gives 6? x = 3
• Trig equation: sin θ = 0.5 → What angle has a sine of 0.5? θ = 30°

🚪 Part 1: Solving Basic Trig Equations

The Three Main Types

Type 1: sin θ = k

If sin θ = 0.5
Think: "Where on the unit circle is the y-value 0.5?"
Answer: θ = 30° (or π/6 radians)

Type 2: cos θ = k

If cos θ = 0.5
Think: "Where is the x-value 0.5?"
Answer: θ = 60° (or π/3 radians)

Type 3: tan θ = k

If tan θ = 1
Think: "Where does y/x = 1?"
Answer: θ = 45° (or π/4 radians)

🎯 Quick Example

Solve: sin θ = √3/2

Step 1: Ask yourself - “Which special angle has this sine value?” Step 2: Check your memory: sin 60° = √3/2 ✓ Step 3: Answer: θ = 60° (or π/3)

🌟 Part 2: Principal Solutions

What Are Principal Solutions?

Think of it like your home address. Just as you have ONE main address, every trig equation has ONE principal (main) solution.

The Principal Zone:

🎯 Example

Find the principal solution of cos θ = -½

Since cos θ is negative, we look in the range 0° to 180°.

• Cos is negative in the second quadrant
• cos 60° = ½, so cos 120° = -½
• Principal solution: θ = 120° (or 2π/3)

🔄 Part 3: General Solutions

Why Do We Need Them?

Here’s a fun fact: Trigonometric functions repeat forever!

Imagine a merry-go-round 🎠:

• Every 360° (or 2π), you’re back where you started
• So if sin 30° = 0.5, then sin 390° = 0.5 too! (That’s 30° + 360°)

The Magic Formulas

For sin θ = k:

θ = nπ + (-1)ⁿ × α
where α is the principal value and n = any integer

For cos θ = k:

θ = 2nπ ± α
where α is the principal value and n = any integer

For tan θ = k:

θ = nπ + α
where α is the principal value and n = any integer

🎯 Example

Find the general solution of sin θ = ½

1. Principal value: α = π/6 (which is 30°)
2. General formula: θ = nπ + (-1)ⁿ × (π/6)

This gives us:

• n = 0: θ = 0 + π/6 = π/6 (30°)
• n = 1: θ = π - π/6 = 5π/6 (150°)
• n = 2: θ = 2π + π/6 = 13π/6 (390°)
• And so on forever!

📏 Part 4: Equations in Intervals

The Bounded Treasure Hunt

Sometimes you don’t need ALL solutions—just the ones within a certain range.

Example: Find all solutions to cos θ = ½ in [0°, 360°)

Step 1: Principal solution = 60° Step 2: Cosine is positive in Q1 and Q4 Step 3: Q1 solution: 60° Step 4: Q4 solution: 360° - 60° = 300°

Answers: θ = 60° and θ = 300°

🎯 Visual Guide

graph TD
    A["cos θ = ½"] --> B{Which quadrants<br/>is cos positive?}
    B --> C["Q1: 60°"]
    B --> D["Q4: 300°"]
    C --> E["Both are in 0° to 360°"]
    D --> E
    E --> F["Final: 60° and 300°"]

🔮 Part 5: The R-Formula Magic

What Is the R-Formula?

Ever tried to add waves together? 🌊 The R-formula lets us combine two trigonometric terms into one beautiful wave!

The Problem: Expressions like 3 sin θ + 4 cos θ are messy. Can we write them as just ONE trig function?

YES! That’s the R-formula!

✨ Part 6: R sin(θ + α) Form

The Formula

We can write:

a sin θ + b cos θ = R sin(θ + α)

Where:

• R = √(a² + b²)
• tan α = b/a

🎯 Step-by-Step Example

Express 3 sin θ + 4 cos θ in the form R sin(θ + α)

Step 1: Find R

R = √(3² + 4²) = √(9 + 16) = √25 = 5

Step 2: Find α

tan α = 4/3
α = tan⁻¹(4/3) ≈ 53.13°

Step 3: Write the answer

3 sin θ + 4 cos θ = 5 sin(θ + 53.13°)

Magic! 🎩 Two terms became one!

🌈 Part 7: R cos(θ + α) Form

The Alternative Formula

Sometimes we prefer the cosine form:

a cos θ + b sin θ = R cos(θ - α)

Or for subtraction:

a cos θ - b sin θ = R cos(θ + α)

Where:

• R = √(a² + b²) (same as before!)
• tan α = b/a

🎯 Example

Express 5 cos θ - 12 sin θ in the form R cos(θ + α)

Step 1: Find R

R = √(5² + 12²) = √(25 + 144) = √169 = 13

Step 2: Find α

tan α = 12/5 = 2.4
α = tan⁻¹(2.4) ≈ 67.38°

Answer:

5 cos θ - 12 sin θ = 13 cos(θ + 67.38°)

🧮 Part 8: Finding R and α Values

The Master Method

Here’s your complete toolkit:

🎯 Practice Example

Express 8 sin θ + 6 cos θ as R sin(θ + α)

Finding R:

R = √(8² + 6²)
R = √(64 + 36)
R = √100
R = 10

Finding α:

tan α = 6/8 = 0.75
α = tan⁻¹(0.75)
α ≈ 36.87° (or 0.6435 radians)

Final Answer:

8 sin θ + 6 cos θ = 10 sin(θ + 36.87°)

🎪 Why Does the R-Formula Matter?

Practical Power

Once you have R sin(θ + α), you can easily:

1. Find the maximum value = R (when sin = 1)
2. Find the minimum value = -R (when sin = -1)
3. Solve equations much more easily!

🎯 Solving with R-Formula

Solve: 3 sin θ + 4 cos θ = 2.5 for 0° ≤ θ < 360°

Step 1: Convert to R-form (we did this!)

5 sin(θ + 53.13°) = 2.5

Step 2: Simplify

sin(θ + 53.13°) = 0.5

Step 3: Solve

θ + 53.13° = 30° or 150°
θ = -23.13° or 96.87°

Step 4: Adjust to interval [0°, 360°)

θ = 336.87° or θ = 96.87°

🗺️ Summary: Your Complete Map

graph TD
    A["Trig Equation"] --> B{What type?}
    B --> C["Basic: sin/cos/tan θ = k"]
    B --> D["Combined: a sin θ + b cos θ = k"]
    C --> E["Find Principal Solution"]
    E --> F["Write General Solution"]
    F --> G["Filter by Interval if needed"]
    D --> H["Use R-Formula"]
    H --> I["Convert to R sin/cos θ + α"]
    I --> J["Solve the simpler equation"]
    J --> G

🏆 Key Takeaways

1. Basic equations → Use special angle values
2. Principal solution → The ONE answer in the standard range
3. General solution → ALL possible answers using + n(period)
4. Interval solutions → Filter answers to fit the given range
5. R-formula → Combines two terms into one using R = √(a² + b²)
6. R and α → R is the amplitude, α is the phase shift

You’ve got this! Every trig equation is just a puzzle waiting to be solved. 🧩