Trigonometry

🔺 Trigonometry: The Art of Triangle Secrets

Imagine you’re a detective. Your job? To find hidden lengths and angles using just a few clues. That’s trigonometry—the ultimate triangle detective toolkit!

🎯 The Big Picture

Think of a right triangle as a pizza slice. One corner has a square angle (90°). The other two corners are your mystery angles. Trigonometry helps you find any missing piece if you know just a little bit.

Why does this matter?

• Pilots use it to navigate the sky ✈️
• Builders use it to make tall towers 🏗️
• Game designers use it to make characters move 🎮

📐 Meet the Triangle Family

Before we dive in, let’s name the parts of our pizza slice:

       /|
      / |
     /  | ← Opposite (across from your angle)
    /   |
   /θ   |
  /_____|
    ↑
 Adjacent (next to your angle)

The longest side (slanted) = Hypotenuse

Simple Memory Trick:

• Hypotenuse = The Hero (always the longest, always opposite the 90°)
• Opposite = Over there (across from your angle θ)
• Adjacent = Attached (touching your angle θ)

🧮 Part 1: Trigonometric Ratios

What Are Ratios?

A ratio is just comparing two things. Like saying “I have 2 apples for every 3 oranges.”

In triangles, we compare sides. These comparisons have special names!

🎭 The Big Three: Sin, Cos, Tan

SOH-CAH-TOA (Your Magic Spell!)

🌟 Example 1: Finding Sin, Cos, Tan

       /|
      / |
     /  | 3
    /   |
   /θ   |
  /_____|
     4

Hypotenuse = 5 (using 3-4-5 rule)

If θ is the bottom-left angle:

• sin θ = Opposite ÷ Hypotenuse = 3 ÷ 5 = 0.6
• cos θ = Adjacent ÷ Hypotenuse = 4 ÷ 5 = 0.8
• tan θ = Opposite ÷ Adjacent = 3 ÷ 4 = 0.75

🔄 The Flip Side: Cosec, Sec, Cot

These are just the upside-down versions!

🌟 Example 2: Using the Same Triangle

From our 3-4-5 triangle:

• cosec θ = 5 ÷ 3 = 1.67
• sec θ = 5 ÷ 4 = 1.25
• cot θ = 4 ÷ 3 = 1.33

📊 Special Angles to Remember Forever!

Pattern Trick: For sin, the values go 0, 1/2, 1/√2, √3/2, 1 (increasing!) For cos, it’s the reverse!

🔗 Part 2: Trigonometric Identities

What’s an Identity?

An identity is a math fact that’s always true. Like saying “I am me” — never wrong!

🏠 The Pythagorean Family

These come from the famous a² + b² = c²

Identity #1: The Foundation

sin²θ + cos²θ = 1

Why it works:

• sin²θ = (O/H)² = O²/H²
• cos²θ = (A/H)² = A²/H²
• Adding them: (O² + A²)/H² = H²/H² = 1 ✓

🌟 Example 3: Using the Foundation

If sin θ = 3/5, find cos θ

1. sin²θ + cos²θ = 1
2. (3/5)² + cos²θ = 1
3. 9/25 + cos²θ = 1
4. cos²θ = 16/25
5. cos θ = 4/5 ✓

Identity #2 & #3: The Cousins

Divide the foundation by cos²θ:

1 + tan²θ = sec²θ

Divide the foundation by sin²θ:

1 + cot²θ = cosec²θ

🔄 Quotient Identities

tan θ = sin θ / cos θ
cot θ = cos θ / sin θ

Think of it: Tan is sin divided by cos. Simple!

🌟 Example 4: Finding Tan

If sin θ = 0.6 and cos θ = 0.8, find tan θ

tan θ = sin θ / cos θ = 0.6 / 0.8 = 0.75 ✓

🪞 Reciprocal Identities

sin θ × cosec θ = 1
cos θ × sec θ = 1
tan θ × cot θ = 1

They’re partners! Multiply them = 1

🎭 Co-function Identities

When angles add up to 90°, they’re complementary:

sin θ = cos(90° - θ)
cos θ = sin(90° - θ)
tan θ = cot(90° - θ)

🌟 Example 5: Co-functions

sin 30° = cos ?

Answer: cos(90° - 30°) = cos 60° = 1/2 ✓

🏔️ Part 3: Heights and Distances

The Real-World Magic!

This is where trigonometry becomes a superpower. You can find heights of buildings, distances across rivers, and more!

📐 Angle of Elevation

When you look UP at something, the angle from horizontal is the angle of elevation.

                    🌟 Top of tower
                   /|
                  / |
                 /  | Height (h)
                /   |
               / θ  |
You 👤_______/______|
    Distance (d)

Formula: tan θ = Height / Distance

🌟 Example 6: Finding Height

You stand 50m from a tower. You look up at 60°. How tall is the tower?

tan 60° = h / 50
√3 = h / 50
h = 50 × √3
h = 50 × 1.732
h ≈ 86.6 meters

📐 Angle of Depression

When you look DOWN from a height, the angle from horizontal is the angle of depression.

You 👤 on cliff ___________
                    \ θ
                     \
                      \
                       \
                        🚗 Car below

Key Insight: Angle of depression from top = Angle of elevation from bottom!

🌟 Example 7: Finding Distance

From a 100m lighthouse, you see a ship at 30° depression. How far is the ship?

tan 30° = 100 / d
1/√3 = 100 / d
d = 100 × √3
d ≈ 173.2 meters

🎯 Two-Position Problems

Sometimes you take measurements from two different spots!

🌟 Example 8: Two Observations

Walking toward a tower, angles of elevation change from 30° to 60°. You walked 100m. Find tower height.

From first position: tan 30° = h / (d + 100)
From second position: tan 60° = h / d

Solving:

1. h = d × tan 60° = d√3
2. h = (d + 100) × tan 30° = (d + 100)/√3

Setting equal: d√3 = (d + 100)/√3 3d = d + 100 2d = 100 d = 50m

Height: h = 50 × √3 ≈ 86.6 meters

🌳 Shadow Problems

The sun creates shadows. The angle of the sun affects shadow length!

🌟 Example 9: Shadow Length

A 6m pole casts a 6m shadow. What’s the sun’s angle?

tan θ = height / shadow
tan θ = 6 / 6 = 1
θ = 45°

🔑 Quick Problem-Solving Steps

1. Draw the picture (always!)
2. Mark what you know (angles, distances)
3. Identify what you need (height? distance?)
4. Pick the right ratio (SOH-CAH-TOA)
5. Solve and check (does the answer make sense?)

📊 Visual Summary: The Trigonometry Flow

graph TD
    A["Right Triangle Problem"] --> B{What do you have?}
    B --> C["Two sides?"]
    B --> D["One side + angle?"]
    C --> E["Use ratios to find angle"]
    D --> F["Use ratios to find missing side"]
    E --> G["sin⁻¹, cos⁻¹, or tan⁻¹"]
    F --> H["SOH-CAH-TOA"]

🎉 You Did It!

You now know:

• ✅ Trigonometric Ratios (sin, cos, tan and their reciprocals)
• ✅ Trigonometric Identities (the always-true formulas)
• ✅ Heights and Distances (real-world applications)

Remember: Every tall building, every navigation system, every video game uses these same simple ratios. You’re now part of a centuries-old mathematical tradition!

“In the world of triangles, no angle is too strange, no height is too tall. With SOH-CAH-TOA by your side, you can measure them all!” 🔺✨