Trigonometry Applications: Your GPS for Triangles 🧭
Imagine you’re a treasure hunter. You have a map, some clues, and mysterious angles. How do you find the hidden gold? Trigonometry is your secret decoder ring!
The Big Picture
Think of trigonometry applications like superhero tools. Each tool solves a different problem:
- Solving triangles = Finding missing pieces of a puzzle
- Inverse trig = Working backwards (like rewinding a movie)
- Elevation/Depression = Looking up at birds or down at fish
- Bearings = Compass directions for pirates and pilots
- Law of Sines = The “fair share” rule for triangles
- Law of Cosines = Pythagoras’s bigger, stronger cousin
- Area formula = Measuring land without a ruler
1. Solving Triangles: The Puzzle Master
What Does “Solving” Mean?
A triangle has 6 parts: 3 sides and 3 angles. “Solving” means finding ALL missing parts when you only know some of them.
The Detective’s Toolkit
| You Know | You Use |
|---|---|
| Two angles + one side | Law of Sines |
| Two sides + angle between | Law of Cosines |
| Two sides + angle opposite | Law of Sines |
| Three sides | Law of Cosines |
Simple Example
You have a triangle where:
- Angle A = 40°
- Angle B = 60°
- Side a = 10
Step 1: Find angle C
C = 180° - 40° - 60° = 80°
Step 2: Use Law of Sines to find other sides
Like a detective, you use what you know to discover what you don’t!
2. Inverse Trig Functions: The Rewind Button
The Problem
Regular trig gives you: angle → ratio
But what if you HAVE the ratio and NEED the angle?
The Solution: Inverse Functions!
| Function | Inverse | What It Does |
|---|---|---|
| sin | sin⁻¹ or arcsin | ratio → angle |
| cos | cos⁻¹ or arccos | ratio → angle |
| tan | tan⁻¹ or arctan | ratio → angle |
Think of It Like This
- sin(30°) = 0.5 → “What’s half of the hypotenuse?” Answer: 0.5
- sin⁻¹(0.5) = 30° → “When is sin equal to half?” Answer: 30°
Simple Example
A ladder leans against a wall. The ladder is 5m long. The base is 3m from the wall. What angle does the ladder make?
cos(θ) = 3/5 = 0.6
θ = cos⁻¹(0.6) = 53.13°
Inverse trig is like asking “which angle created this ratio?”
3. Angles of Elevation and Depression
The Concept
Imagine you’re standing flat on the ground, looking straight ahead. This is your horizontal line.
- Elevation = You look UP (like at a kite)
- Depression = You look DOWN (like at a boat from a cliff)
graph TD A["👀 Your Eye"] -->|Look UP| B["🪁 Kite - Elevation"] A -->|Horizontal Line| C["➡️ Straight Ahead"] A -->|Look DOWN| D["🚤 Boat - Depression"]
The Magic Trick
Depression angle = Elevation angle (they’re alternate angles!)
From a cliff, you look DOWN at a boat at 25°. From the boat looking UP at you? Also 25°!
Simple Example
You stand 50m from a tree. You look UP at the top at 32°.
tan(32°) = height/50
height = 50 × tan(32°)
height = 50 × 0.625 = 31.25m
The tree is about 31 meters tall!
4. Bearings and Navigation: The Pirate’s Way
What Are Bearings?
Bearings are directions measured from NORTH, going clockwise, as a 3-digit number.
graph TD N["North 000°"] --> E["East 090°"] E --> S["South 180°"] S --> W["West 270°"] W --> N
Key Rules
- Always start from NORTH
- Always go CLOCKWISE
- Always write 3 digits (045°, not 45°)
Common Bearings
| Direction | Bearing |
|---|---|
| North | 000° |
| East | 090° |
| South | 180° |
| West | 270° |
| NE | 045° |
| SW | 225° |
Simple Example
A ship sails 10km on bearing 060°, then 8km on bearing 150°. How far from start?
Draw a triangle! Use the Law of Cosines:
- The angle between paths = 150° - 60° = 90°
distance² = 10² + 8² = 164
distance = √164 = 12.8km
The ship is about 13km from where it started!
5. Law of Sines: The Fair Share Rule
The Formula
In ANY triangle:
a b c
───── = ───── = ─────
sin A sin B sin C
Small letters = sides. Big letters = angles opposite to those sides.
When to Use It
✅ You have an angle and its opposite side, plus one more piece ✅ AAS (two angles, one side) ✅ ASA (angle, side, angle) ✅ SSA (two sides, angle opposite one) - careful, this can have 2 answers!
Simple Example
Triangle with:
- Angle A = 45°
- Angle B = 70°
- Side a = 20
Find side b:
20 b
───── = ─────
sin45° sin70°
b = 20 × sin70° ÷ sin45°
b = 20 × 0.940 ÷ 0.707
b = 26.6
Side b is about 26.6 units long!
6. Law of Cosines: The Super Pythagoras
Why We Need It
Pythagoras only works for RIGHT triangles. But the Law of Cosines works for ANY triangle!
The Formula
c² = a² + b² - 2ab × cos(C)
Notice: if C = 90°, then cos(90°) = 0, and we get c² = a² + b² … that’s Pythagoras!
When to Use It
✅ SAS (two sides and the angle BETWEEN them) ✅ SSS (all three sides, finding an angle)
Simple Example: Finding a Side
Two sides are 7 and 9. The angle between them is 65°.
c² = 7² + 9² - 2(7)(9)cos(65°)
c² = 49 + 81 - 126(0.423)
c² = 130 - 53.3 = 76.7
c = √76.7 = 8.76
The third side is about 8.8 units!
Simple Example: Finding an Angle
Three sides: a = 8, b = 6, c = 10
cos(C) = (8² + 6² - 10²) ÷ (2 × 8 × 6)
cos(C) = (64 + 36 - 100) ÷ 96
cos(C) = 0 ÷ 96 = 0
C = cos⁻¹(0) = 90°
It’s a right triangle!
7. Area Using Sine Formula: No Height Needed!
The Problem
The normal area formula needs HEIGHT:
Area = ½ × base × height
But what if you don’t know the height?
The Magic Formula
Area = ½ × a × b × sin(C)
Where a and b are two sides, and C is the angle BETWEEN them!
Why It Works
graph TD A["Side a"] --> B["Angle C between them"] C["Side b"] --> B B --> D["Height = b × sin C"] D --> E["Area = ½ × a × height"] E --> F["Area = ½ × a × b × sin C"]
Simple Example
Triangle with:
- Side a = 12
- Side b = 8
- Angle C = 40° (between a and b)
Area = ½ × 12 × 8 × sin(40°)
Area = ½ × 12 × 8 × 0.643
Area = 30.9 square units
The area is about 31 square units!
Putting It All Together: The Treasure Hunt
Let’s solve a real problem using everything!
Problem: A treasure hunter stands at point A. He sees a tower at bearing 045° that is 100m away. The treasure is at bearing 120° from A, and from the tower, the treasure is due south. How far is the treasure from A?
Solution:
- Draw the situation
- Angle at A = 120° - 45° = 75°
- The treasure being “due south” of tower = bearing 180°
- Angle at tower = 180° - 45° = 135°
- Angle at treasure = 180° - 75° - 135° = -30°… wait!
In bearing problems, we need to be careful about interior vs exterior angles!
The interior angle at the tower = 180° - 135° = 45° Angle at treasure = 180° - 75° - 45° = 60°
Using Law of Sines:
AT 100
────── = ────────
sin(45°) sin(60°)
AT = 100 × sin(45°) ÷ sin(60°)
AT = 100 × 0.707 ÷ 0.866
AT = 81.6m
The treasure is about 82m from the hunter!
Quick Reference Summary
| Tool | When to Use | Formula |
|---|---|---|
| Law of Sines | AAS, ASA, SSA | a/sinA = b/sinB |
| Law of Cosines | SAS, SSS | c² = a² + b² - 2ab·cosC |
| Area | Have 2 sides + included angle | ½ab·sinC |
| Inverse Trig | Know ratio, need angle | sin⁻¹, cos⁻¹, tan⁻¹ |
| Elevation | Looking UP from horizontal | Use tan usually |
| Depression | Looking DOWN from horizontal | Same angle as elevation from below |
| Bearings | Navigation problems | 3 digits, from N, clockwise |
You Did It! 🎉
You now have a complete toolkit for solving ANY triangle problem:
- Identify what you know (sides? angles?)
- Choose the right tool (Law of Sines? Cosines? Inverse trig?)
- Solve step by step
- Check - do your answers make sense?
Remember: Every expert was once a beginner. Keep practicing, and these tools will become second nature!