Inverse Formulas and Graphs

🎯 Inverse Trig Functions: The “Undo” Button for Trigonometry

The Big Picture: Finding Your Way Back Home

Imagine you’re playing hide-and-seek. Your friend ran away at a certain angle and ended up somewhere. Now you know where they are, but you want to figure out which direction they ran. That’s exactly what inverse trig functions do!

Regular trig functions (sin, cos, tan) take an angle and tell you a ratio. Inverse trig functions take a ratio and tell you the angle.

It’s like having an “undo” button! 🔄

🧙‍♂️ Meet Our Heroes: The Inverse Functions

Think of it this way:

• sin(30°) = 0.5 → “What’s the sine of 30°? It’s 0.5!”
• arcsin(0.5) = 30° → “What angle has sine = 0.5? It’s 30°!”

📐 The 2 Arctan x Formulas

These formulas help you work with double angles using arctan. Think of them as “shortcuts” to avoid messy calculations!

Formula 1: Two Arctans Adding Up

arctan(a) + arctan(b) = arctan((a + b)/(1 - ab)) when ab < 1

Example: What is arctan(1/2) + arctan(1/3)?

Using our formula:

• a = 1/2, b = 1/3
• (a + b)/(1 - ab) = (1/2 + 1/3)/(1 - 1/6) = (5/6)/(5/6) = 1
• So arctan(1/2) + arctan(1/3) = arctan(1) = π/4 or 45°

Formula 2: The Subtraction Version

arctan(a) - arctan(b) = arctan((a - b)/(1 + ab))

Example: arctan(3) - arctan(1/2)

• (3 - 1/2)/(1 + 3/2) = (5/2)/(5/2) = 1
• Answer: π/4 or 45°

🔢 The 3 Arctan x Formulas

When you have three arctan terms to combine, here’s the magic formula:

arctan(a) + arctan(b) + arctan© = arctan((a + b + c - abc)/(1 - ab - bc - ca))

Example: arctan(1) + arctan(1/2) + arctan(1/3)

Let’s solve step by step:

• a = 1, b = 1/2, c = 1/3
• Numerator: 1 + 1/2 + 1/3 - (1)(1/2)(1/3) = 11/6 - 1/6 = 10/6 = 5/3
• Denominator: 1 - (1)(1/2) - (1/2)(1/3) - (1/3)(1) = 1 - 1/2 - 1/6 - 1/3 = 0

Wait! Denominator is 0? That means the result is π/2 or 90°! 🎉

🤝 Arcsin-Arccos Relations: Best Friends Forever!

Here’s a beautiful secret: arcsin and arccos are like two friends who always balance each other out!

The Golden Rule

arcsin(x) + arccos(x) = π/2 (for -1 ≤ x ≤ 1)

This means: Whatever angle arcsin gives you, arccos gives you its “complement” (what you need to reach 90°).

Example:

• arcsin(1/2) = 30° = π/6
• arccos(1/2) = 60° = π/3
• Add them: 30° + 60° = 90° = π/2 ✓

More Helpful Relations

🔍 Solving Inverse Trig Equations

Solving these equations is like being a detective! Let’s learn the strategy.

Step-by-Step Process

1. Isolate the inverse trig function
2. Apply the regular trig function to both sides
3. Check if your answer is in the valid range

Example 1: Simple Equation

Solve: arcsin(x) = π/6

Solution:

• Take sine of both sides: sin(arcsin(x)) = sin(π/6)
• This gives: x = 1/2 ✓

Example 2: With Addition

Solve: arcsin(x) + arccos(x) = π/2

• We know this is ALWAYS true for -1 ≤ x ≤ 1
• So x can be any value from -1 to 1!

Example 3: Finding Unknown

Solve: 2·arctan(x) = π/4

• arctan(x) = π/8
• x = tan(π/8) ≈ 0.414

📊 Graph of Arcsin (y = sin⁻¹x)

graph TD
    A["Domain: -1 ≤ x ≤ 1"] --> B["Range: -π/2 to π/2"]
    B --> C["Passes through origin #40;0,0#41;"]
    C --> D["Increasing function"]
    D --> E["S-shaped curve"]

Key Features:

• Domain: [-1, 1] (You can only find arcsin of numbers between -1 and 1)
• Range: [-π/2, π/2] or [-90°, 90°]
• Shape: Looks like a stretched “S” lying on its side
• Special Points:
  • arcsin(-1) = -π/2 = -90°
  • arcsin(0) = 0
  • arcsin(1) = π/2 = 90°

Visual Analogy: Imagine a snake stretching between two walls at x = -1 and x = 1!

📊 Graph of Arccos (y = cos⁻¹x)

graph TD
    A["Domain: -1 ≤ x ≤ 1"] --> B["Range: 0 to π"]
    B --> C["Starts at #40;−1, π#41;"]
    C --> D["Decreasing function"]
    D --> E["Ends at #40;1, 0#41;"]

Key Features:

• Domain: [-1, 1]
• Range: [0, π] or [0°, 180°]
• Shape: Like a slide going down from left to right
• Special Points:
  • arccos(-1) = π = 180°
  • arccos(0) = π/2 = 90°
  • arccos(1) = 0

Why decreasing? As x increases, the angle needed to get that cosine value decreases!

📊 Graph of Arctan (y = tan⁻¹x)

graph TD
    A["Domain: All real numbers!"] --> B["Range: -π/2 to π/2"]
    B --> C["Has horizontal asymptotes"]
    C --> D["Passes through #40;0, 0#41;"]
    D --> E["Always increasing"]

Key Features:

• Domain: (-∞, ∞) - Any real number works!
• Range: (-π/2, π/2) - but never reaches the edges
• Asymptotes:
  • As x → ∞, arctan(x) → π/2
  • As x → -∞, arctan(x) → -π/2
• Special Points:
  • arctan(0) = 0
  • arctan(1) = π/4 = 45°
  • arctan(-1) = -π/4 = -45°

Visual Analogy: Like a road that goes forever but can never climb past a certain height!

📊 Graphs of Other Inverse Functions

Arccot (y = cot⁻¹x)

Example: arccot(1) = π/4 = 45°

Arcsec (y = sec⁻¹x)

Example: arcsec(2) = π/3 = 60° (because sec(60°) = 2)

Arccsc (y = csc⁻¹x)

Example: arccsc(2) = π/6 = 30° (because csc(30°) = 2)

🎯 Quick Reference: All Graphs at a Glance

🌟 Why This Matters

Inverse trig functions are everywhere!

• Navigation: GPS uses them to calculate directions
• Engineering: Designing ramps and slopes
• Physics: Finding angles in projectile motion
• Computer Graphics: Rotating objects on screen

You now have the “undo” button for trigonometry. When someone gives you a ratio, you can find the angle. That’s real power! 💪

🎮 Key Takeaways

1. Inverse trig functions reverse regular trig functions - they find angles from ratios
2. 2 and 3 arctan formulas help combine multiple arctans into one
3. arcsin + arccos = π/2 is the golden relationship
4. Each graph has specific domain and range - memorize them!
5. Arctan is the friendliest - accepts any real number
6. Arcsec and arccsc have gaps - they don’t accept values between -1 and 1