Other Trig Graphs

🎢 The Wild Roller Coasters of Trigonometry

Graphing Tangent, Cotangent, Secant & Cosecant Functions

🎡 Our Magical Metaphor: The Infinite Playground

Imagine a playground with special rides. Some rides go up and down smoothly (like sine and cosine). But today, we meet the wild rides—the ones that zoom up to the sky and dive down to the center of the Earth!

These are our tangent, cotangent, secant, and cosecant functions. They have something special: invisible walls called asymptotes where the ride can never go.

🎯 The Tangent Function: The Fearless Climber

What is Tangent?

Think of a ladder leaning against a wall. As you tilt the ladder more and more, it reaches higher and higher up the wall—until suddenly, it’s standing straight up and reaches infinity!

Tangent = sin(x) ÷ cos(x)

When cos(x) = 0, we can’t divide by zero. That’s where the invisible walls appear!

📊 The Tangent Graph

        |           /|          /|
        |          / |         / |
        |         /  |        /  |
   -----+--------/---+-------/---+-----
        |       /    |      /    |
        |      /     |     /     |
        |     /      |    /      |
      -π/2    0     π/2   π    3π/2

🔑 Key Features of y = tan(x)

🎨 Example: Graphing y = tan(x)

1. Draw vertical dashed lines at x = -π/2, π/2, 3π/2 (asymptotes)
2. Plot the center point (0, 0)
3. The curve rises from bottom-left to top-right between asymptotes
4. It never touches the dashed lines!

🔄 The Cotangent Function: Tangent’s Mirror Twin

What is Cotangent?

Cotangent is like tangent looking in a mirror while doing a flip!

Cotangent = cos(x) ÷ sin(x) = 1 ÷ tan(x)

Now the invisible walls appear when sin(x) = 0 instead!

📊 The Cotangent Graph

        |\          |\          |
        | \         | \         |
        |  \        |  \        |
   -----+---\-------+---\-------+-----
        |    \      |    \      |
        |     \     |     \     |
        |      \|   |      \|   |
        0      π    2π     3π

🔑 Key Features of y = cot(x)

🎨 Example: Graphing y = cot(x)

1. Draw asymptotes at x = 0, π, 2π
2. Plot (π/2, 0) as center point
3. The curve falls from top-left to bottom-right (opposite of tangent!)
4. Repeat the pattern

🦸 The Secant Function: Cosine’s Superhero Partner

What is Secant?

Secant is cosine with superpowers—it’s the reciprocal!

Secant = 1 ÷ cos(x)

When cosine is small, secant becomes HUGE! When cosine is zero, secant flies off to infinity!

📊 The Secant Graph

      U          U          U
     / \        / \        / \
    /   \      /   \      /   \
   /     \    /     \    /     \
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \     /    \     /    \     /
    \   /      \   /      \   /
     \ /        \ /        \ /
      ∩          ∩          ∩

It looks like a series of U shapes and upside-down U shapes!

🔑 Key Features of y = sec(x)

🎨 Example: Graphing y = sec(x)

1. First sketch y = cos(x) lightly (as a guide)
2. Draw asymptotes where cos(x) = 0
3. Where cos(x) = 1, sec(x) = 1 (touch point)
4. Where cos(x) = -1, sec(x) = -1 (touch point)
5. Draw U curves opening away from x-axis

🛡️ The Cosecant Function: Sine’s Guardian

What is Cosecant?

Just like secant guards cosine, cosecant guards sine!

Cosecant = 1 ÷ sin(x)

📊 The Cosecant Graph

  U          U          U
 / \        / \        / \
/   \      /   \      /   \
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\   /      \   /      \   /
 \ /        \ /        \ /
  ∩          ∩          ∩
   0    π    2π   3π   4π

🔑 Key Features of y = csc(x)

🎨 Example: Graphing y = csc(x)

1. First sketch y = sin(x) lightly
2. Draw asymptotes at x = 0, π, 2π, 3π…
3. Where sin(x) = 1, csc(x) = 1
4. Where sin(x) = -1, csc(x) = -1
5. Draw U curves opening away from the x-axis

🚧 Asymptote Analysis: The Invisible Walls

What is an Asymptote?

Imagine chasing a rainbow 🌈. No matter how fast you run, you never reach it. An asymptote is like that rainbow—a line the graph gets closer and closer to but never touches or crosses.

The Asymptote Rules

graph TD
    A["Find Asymptotes"] --> B{Which Function?}
    B --> C["tan or sec"]
    B --> D["cot or csc"]
    C --> E["Asymptotes where<br/>cos x = 0"]
    D --> F["Asymptotes where<br/>sin x = 0"]
    E --> G["x = π/2 + nπ"]
    F --> H["x = nπ"]

📋 Asymptote Quick Reference

🎯 Example: Finding Asymptotes

Question: Find the asymptotes of y = tan(2x)

Solution:

1. Tangent has asymptotes when cos(2x) = 0
2. cos(θ) = 0 when θ = π/2 + nπ
3. So 2x = π/2 + nπ
4. x = π/4 + nπ/2

The asymptotes are at x = π/4, 3π/4, 5π/4, …

🎯 Domain and Range from Graphs

Understanding Domain and Range

Think of domain as the horizontal road your graph can travel on. Think of range as the vertical space where your graph can live.

📊 The Complete Picture

graph TD
    A["Reading Domain & Range"] --> B["DOMAIN: Look left to right"]
    A --> C["RANGE: Look up and down"]
    B --> D["Where can x go?"]
    C --> E["Where can y reach?"]
    D --> F["Skip the asymptotes!"]
    E --> G["Check gaps in y-values"]

📋 Domain and Range Summary

🎯 Example: Reading from a Graph

Looking at y = sec(x):

1. Domain: Look horizontally

   • There are vertical asymptotes at π/2, 3π/2, etc.
   • Domain = all real numbers EXCEPT these points

2. Range: Look vertically

   • The graph never enters the band between -1 and 1
   • Range = (-∞, -1] ∪ [1, +∞)

🏆 Summary: Your Trig Graph Superpowers

The Function Families

Team Tangent (tan & cot):

• 🎢 Go from -∞ to +∞
• 📏 Period of π
• 🎯 Range = all real numbers

Team Secant (sec & csc):

• 🎪 Make U-shaped curves
• 📏 Period of 2π
• 🎯 Range = y ≤ -1 or y ≥ 1 (avoid the middle!)

Golden Rules

1. ✅ Always find asymptotes first (the invisible walls)
2. ✅ Sketch the “parent” function (cos for sec, sin for csc)
3. ✅ Period of tan/cot = π | Period of sec/csc = 2π
4. ✅ Tan/cot reach all y-values | Sec/csc skip between -1 and 1

🌟 You Did It!

You now understand the wild roller coasters of trigonometry! These graphs might look scary at first, but remember:

• Asymptotes are just invisible walls where we can’t divide by zero
• Sec and csc are just reciprocals of cos and sin
• Every graph has a repeating pattern (period)

Keep practicing, and soon these graphs will feel as natural as riding a bike! 🚴‍♀️