Inverse Functions Basics

🔓 Unlocking the Mystery: Inverse Trig Functions

The Backwards Magic Trick

Imagine you have a magic box. You put in a number, and it gives you an angle. That’s what regular trig functions do!

But what if you want to go BACKWARDS?

You have the answer… and you want to find the angle that made it. That’s where inverse trig functions come in!

🤔 Why Do We Need Inverse Functions?

The Lost Angle Problem

Picture this: You’re a detective. Someone tells you:

“The sine of the mystery angle is 0.5”

Your job? Find that mystery angle!

sin(?) = 0.5

Without inverse functions, you’re stuck guessing. With them? You solve it instantly!

? = arcsin(0.5) = 30°

Real Life Example

A ladder leans against a wall. You know:

• The ladder is 10 feet long
• It reaches 5 feet up the wall

What angle does it make with the ground?

sin(angle) = 5/10 = 0.5
angle = arcsin(0.5) = 30°

Inverse functions let you work BACKWARDS from an answer to find the angle!

🎯 The Big Three Inverse Functions

Think of these as “undo buttons” for sine, cosine, and tangent:

📐 Inverse Sine (arcsin)

What Is It?

arcsin answers: “What angle has this sine value?”

Simple Story

You throw a ball. It goes up at some angle.

The “vertical part” of its speed divided by total speed = 0.707

What’s the throwing angle?

arcsin(0.707) = 45°

Quick Examples

Visual Flow

graph TD
    A["You have: sin value"] --> B["Use arcsin"]
    B --> C["Get: the angle!"]
    D["arcsin#40;0.5#41;"] --> E["= 30°"]

📐 Inverse Cosine (arccos)

What Is It?

arccos answers: “What angle has this cosine value?”

Simple Story

You’re watching a shadow. At noon, there’s almost no shadow (sun overhead).

The shadow length divided by pole height = 0.5

What angle is the sun from straight up?

arccos(0.5) = 60°

Quick Examples

Visual Flow

graph TD
    A["You have: cos value"] --> B["Use arccos"]
    B --> C["Get: the angle!"]
    D["arccos#40;0.5#41;"] --> E["= 60°"]

📐 Inverse Tangent (arctan)

What Is It?

arctan answers: “What angle has this tangent value?”

Simple Story

You’re climbing a ramp. For every 1 meter you walk forward, you go 1 meter up.

Rise over run = 1/1 = 1

What’s the ramp angle?

arctan(1) = 45°

Quick Examples

Visual Flow

graph TD
    A["You have: tan value"] --> B["Use arctan"]
    B --> C["Get: the angle!"]
    D["arctan#40;1#41;"] --> E["= 45°"]

🎭 The Problem: Too Many Answers!

Wait… There’s a Twist!

Here’s something tricky. If sin(30°) = 0.5…

Then sin(150°) ALSO = 0.5!

And sin(390°) = 0.5 too! And sin(-210°) = 0.5!

😱 There are INFINITE angles with the same sine value!

The Challenge

If arcsin(0.5) has infinite answers, how does your calculator know which one to give?

This is why we need PRINCIPAL VALUES!

👑 Principal Value Branch

The One True Answer

To solve the “too many answers” problem, mathematicians picked ONE special answer for each inverse function.

This special answer is called the PRINCIPAL VALUE.

Think of It Like This

Imagine a library with infinite copies of the same book on infinite shelves.

When you ask for “the book,” the librarian always goes to ONE specific shelf - the “main” shelf.

That’s the principal value branch!

The Principal Ranges

Why These Ranges?

• arcsin: Goes from bottom (−90°) to top (90°) - covers one “bump” of sine
• arccos: Goes from right (0°) to left (180°) - covers one “bump” of cosine
• arctan: Goes from bottom (−90°) to top (90°) - covers the “S-curve” of tangent

🚧 Restricted Domains Explained

Why Can’t We Use All Values?

For a function to have an inverse, it must be ONE-TO-ONE.

This means: each input gives exactly ONE output.

The Problem with Trig Functions

Regular sine, cosine, and tangent are NOT one-to-one:

• sin(30°) = sin(150°) = 0.5 (same output, different inputs!)

The Solution: Restrict the Domain!

We LIMIT which angles we look at:

graph TD
    A["Full Sine Wave"] --> B["Cut out ONE piece"]
    B --> C[Now it's one-to-one!]
    C --> D["Can have inverse!"]

Restricted Domains for Each

For arcsin to work:

• We only use sine from −90° to 90°
• In this range, each sine value happens only ONCE

For arccos to work:

• We only use cosine from 0° to 180°
• In this range, each cosine value happens only ONCE

For arctan to work:

• We only use tangent from −90° to 90° (not including the endpoints)
• In this range, each tangent value happens only ONCE

Visual Summary

🎮 Putting It All Together

The Complete Picture

graph TD
    A["INVERSE TRIG"] --> B["arcsin"]
    A --> C["arccos"]
    A --> D["arctan"]

    B --> E["Input: -1 to 1"]
    B --> F["Output: -90° to 90°"]

    C --> G["Input: -1 to 1"]
    C --> H["Output: 0° to 180°"]

    D --> I["Input: any number"]
    D --> J["Output: -90° to 90°"]

Quick Memory Trick

🎵 “Arc-SIGN goes UP and DOWN” (−90° to 90°)

• Like a sign pointing up or down

🎵 “Arc-COS goes LEFT to RIGHT” (0° to 180°)

• Like the sun’s path across the sky

🎵 “Arc-TAN is like arc-SIN” (−90° to 90°)

• They’re twins!

🌟 Final Examples

Example 1: Finding an Angle

Problem: A right triangle has opposite side = 3, hypotenuse = 6. Find the angle.

Solution:

sin(θ) = opposite/hypotenuse = 3/6 = 0.5
θ = arcsin(0.5) = 30°

Example 2: Using Principal Value

Problem: Find arccos(−0.5)

Solution:

• cos(120°) = −0.5
• cos(240°) = −0.5 too!
• But arccos only gives 0° to 180°
• Answer: 120° ✓

Example 3: Why Domain Matters

Problem: Can you find arcsin(2)?

Solution:

• Sine values only go from −1 to 1
• There’s NO angle whose sine is 2
• arcsin(2) = UNDEFINED ❌

🎯 Key Takeaways

1. Inverse trig functions find ANGLES when you know the ratio
2. Principal values give ONE specific answer
3. Restricted domains make the functions one-to-one
4. arcsin & arctan: Output from −90° to 90°
5. arccos: Output from 0° to 180°
6. arcsin & arccos only accept inputs from −1 to 1
7. arctan accepts ANY real number

You’ve unlocked the power to work backwards! 🔓✨