Domain and Range

🎢 The Theme Park of Trigonometry: Domain, Range & Function Properties

Imagine trigonometry functions as rides at a theme park. Each ride has rules about who can get on (domain), how high or low you’ll go (range), and special patterns that make each ride unique!

🎡 The Six Trig Functions: Meet the Rides!

Think of our six trig functions as six different rides at “Trig Land”:

• Sine (sin) - The Smooth Wave Coaster 🌊
• Cosine (cos) - The Twin Wave Coaster 🌊
• Tangent (tan) - The Vertical Drop Tower 🗼
• Cosecant (csc) - The Inverted Wave 🔄
• Secant (sec) - The Inverted Twin 🔄
• Cotangent (cot) - The Reverse Drop Tower 🗼

🎫 Domain: Who Can Ride?

Domain = Which numbers (angles) are allowed to go INTO the function

Think of it like height requirements at theme parks. Some rides let EVERYONE on. Others have restrictions!

Rides Open to EVERYONE (All Real Numbers)

Sin(x) and Cos(x) - The Smooth Waves

• Domain: All real numbers (-∞, +∞)
• Any angle works! Put in 0°, 90°, 1000°, -500°… they all work!

sin(0) = 0 ✓
sin(90°) = 1 ✓
sin(12345°) = some number ✓
cos(anything) = works! ✓

Why? On a circle, you can spin around forever in either direction. There’s no “forbidden” angle!

Rides with “CLOSED” Signs (Restrictions)

Tan(x) and Sec(x) - The Vertical Drops

• Domain: All real numbers EXCEPT x = 90° + (180° × n)
• In radians: x ≠ π/2 + nπ (where n is any integer)

tan(0°) = 0 ✓
tan(45°) = 1 ✓
tan(90°) = UNDEFINED! 🚫 (ride closed!)
tan(270°) = UNDEFINED! 🚫

Why banned at 90°, 270°, etc.? At these angles, cosine = 0, and tan = sin/cos. Division by zero = ride malfunction! 💥

Csc(x) and Cot(x) - The Other Restricted Rides

• Domain: All real numbers EXCEPT x = 180° × n
• In radians: x ≠ nπ (where n is any integer)

csc(90°) = 1 ✓
csc(0°) = UNDEFINED! 🚫
csc(180°) = UNDEFINED! 🚫

Why? These involve dividing by sine. When sin = 0 (at 0°, 180°, 360°…), you can’t divide!

📊 Domain Summary Table

🎢 Range: How High Will You Go?

Range = What numbers (outputs) can possibly come OUT of the function

Like knowing the highest and lowest points of a roller coaster!

The Bounded Rides: Sin and Cos

Sin(x) and Cos(x)

• Range: [-1, 1] (stuck between -1 and 1, forever!)

sin(30°) = 0.5 ✓ (inside the range)
sin(90°) = 1 ✓ (maximum height!)
sin(270°) = -1 ✓ (minimum depth!)

Real Life Example: A wave in the ocean goes up 1 meter, down 1 meter, up 1 meter… never goes to 10 meters!

graph TD
    A["sin#40;x#41; output"] --> B["Maximum: 1"]
    A --> C["Minimum: -1"]
    B --> D["Occurs at 90°, 450°..."]
    C --> E["Occurs at 270°, -90°..."]

The Unbounded But Restricted: Sec and Csc

Sec(x) and Csc(x)

• Range: (-∞, -1] ∪ [1, +∞)
• Translation: Can be any number ≤ -1 OR any number ≥ 1
• But NEVER between -1 and 1!

sec(0°) = 1 ✓
sec(60°) = 2 ✓
csc(30°) = 2 ✓
csc(45°) = 1.414... ✓
Can sec = 0.5? NEVER! 🚫

Why this gap? Since sec = 1/cos and |cos| ≤ 1, then |sec| ≥ 1. You can’t make 1/(small number) be a small number!

The Wild Rides: Tan and Cot

Tan(x) and Cot(x)

• Range: All real numbers (-∞, +∞)

tan(0°) = 0
tan(45°) = 1
tan(60°) = 1.732...
tan(89°) = 57.29...
tan(89.9°) = 572.96...
tan(89.99°) = 5729.58...

As you get closer to 90°, tangent shoots to infinity! Then jumps to negative infinity on the other side!

🔄 Periodicity: The Repeating Pattern

Periodicity = How long until the function starts repeating itself

Like a merry-go-round that spins around and returns to the same spot!

Short Cycles: Period = 360° (or 2π)

Sin(x), Cos(x), Sec(x), Csc(x) - One full circle!

sin(0°) = 0
sin(360°) = 0 (same!)
sin(720°) = 0 (same again!)
sin(x) = sin(x + 360°) ALWAYS!

Shorter Cycles: Period = 180° (or π)

Tan(x) and Cot(x) - Half a circle!

tan(45°) = 1
tan(225°) = 1 (same after 180°!)
tan(x) = tan(x + 180°) ALWAYS!

Why shorter? Tangent already “finishes its story” in half a circle because it goes from -∞ to +∞ in just 180°!

graph TD
    A["Periodicity"] --> B["Long Period: 360°"]
    A --> C["Short Period: 180°"]
    B --> D["sin, cos, sec, csc"]
    C --> E["tan, cot"]

🪞 Odd and Even: The Mirror Test

Even Function = Same output for +x and -x (symmetric around y-axis) Odd Function = Opposite output for +x and -x (symmetric around origin)

Even Functions: cos and sec

cos(-x) = cos(x) - It’s a mirror!

cos(30°) = 0.866...
cos(-30°) = 0.866... (same!)

sec(60°) = 2
sec(-60°) = 2 (same!)

Think of it: If you walk 30° left or 30° right from the starting line, you end up at the same height!

Odd Functions: sin, tan, cot, csc

sin(-x) = -sin(x) - Opposite!

sin(30°) = 0.5
sin(-30°) = -0.5 (opposite!)

tan(45°) = 1
tan(-45°) = -1 (opposite!)

cot(60°) = 0.577...
cot(-60°) = -0.577... (opposite!)

csc(30°) = 2
csc(-30°) = -2 (opposite!)

Memory Trick:

• Even (like cosine): The “E” stands for “Equal on both sides”
• Odd (like sine): The “O” stands for “Opposite signs”

📏 Bounded vs Unbounded: Fenced or Free?

Bounded Functions: Sin and Cos (The Fenced Playground)

• Bounded means stuck between fences
• |sin(x)| ≤ 1 and |cos(x)| ≤ 1 ALWAYS
• No matter what crazy angle you try, the answer stays between -1 and 1

sin(1000000°) = some number between -1 and 1
cos(-9999999°) = some number between -1 and 1

Unbounded Functions: The Rest (Free to Roam!)

Tan, Cot, Sec, Csc can grow infinitely large or small!

tan(89.999°) ≈ 57,296 (huge!)
sec(89.999°) ≈ 57,296 (huge!)
csc(0.001°) ≈ 57,296 (huge!)

Visual:

graph TD
    A["Bounded: Sin & Cos"] --> B["Always between -1 and 1"]
    C["Unbounded: Tan, Cot, Sec, Csc"] --> D["Can be ANY size!"]
    C --> E["Shoot to ∞ near banned angles"]

📈 Monotonicity: Going Up or Down?

Monotonicity = Is the function climbing up, sliding down, or switching?

Sin(x): The Wave Climber

In one period [0°, 360°]:

• Increasing (going up): 0° to 90°, and 270° to 360°
• Decreasing (going down): 90° to 270°

From 0° to 90°: sin grows 0 → 1 (climbing!)
From 90° to 270°: sin falls 1 → -1 (sliding!)
From 270° to 360°: sin grows -1 → 0 (climbing!)

Cos(x): The Early Bird Wave

In one period [0°, 360°]:

• Decreasing: 0° to 180°
• Increasing: 180° to 360°

cos(0°) = 1 (starts at top!)
cos(180°) = -1 (bottom)
cos(360°) = 1 (back to top)

Tan(x): The Eternal Climber

In each period, tan is ALWAYS increasing!

On (-90°, 90°):

tan(-89°) ≈ -57 (very negative)
tan(0°) = 0
tan(89°) ≈ 57 (very positive)

Always going UP within each allowed interval!

Cot(x): The Eternal Slider

In each period, cot is ALWAYS decreasing!

On (0°, 180°):

cot(1°) ≈ 57 (very positive)
cot(90°) = 0
cot(179°) ≈ -57 (very negative)

Always going DOWN!

🎯 Quick Reference Card

🌟 The Big Picture

Think of our theme park one more time:

1. Domain = Which tickets (angles) get you in
2. Range = How high/low the ride goes
3. Periodicity = How long until the ride repeats
4. Even/Odd = Mirror symmetry behavior
5. Bounded = Is there a safety fence?
6. Monotonicity = Is it climbing or dropping right now?

Now you know every ride in Trig Land! 🎢🎡🎠

Remember: Sine and Cosine are the “safe” functions - they let everyone in and keep everyone between -1 and 1. The others are wilder, with some restrictions but infinite possibilities!