Lesson 3: Trigonometric Graphs – Sine and Cosine

👨🏽‍🏫 Introduction

Have you ever watched waves at the beach or seen a fan spinning and wondered how their movements can be measured? 🌊🌀 The rise and fall of such movements follow a smooth, repeating curve—just like the sine and cosine graphs you’ll learn in this lesson!

Understanding these graphs helps you visualize how angles change and how wave-like behavior works in math, sound, engineering, and even electricity! ⚡🎵

• 📈 Understand the shapes and patterns of sine and cosine graphs
• 📏 Learn key features like amplitude, period, and axis shifts
• 🧭 Apply trig graphs to real-world cycles like tides, rotations, and oscillations

🧠 Key Concepts

📌 1. Sine Graph – y = sin(x)

• The sine graph starts at 0, rises to 1 at 90°, back to 0 at 180°, -1 at 270°, and back to 0 at 360°.
• It is a wave that repeats every 360°.

📌 2. Cosine Graph – y = cos(x)

• The cosine graph starts at 1, drops to 0 at 90°, -1 at 180°, back to 0 at 270°, and 1 again at 360°.
• It’s the same wave as sine, just shifted to the left by 90°.

📌 3. Key Features of Both Graphs

• Amplitude: The height from the middle to the peak (default is 1)
• Period: How long it takes to complete one full cycle (default is 360° or radians)
• Range: Values go from -1 to 1
• Axis: The center horizontal line (y = 0 for basic sine/cosine)

🔍 Worked Examples

1. 1. Example 1 – Find sin(90°):
      From the sine graph, at 90° the value is:
      
      🟢 Answer: 1

1. 1. Example 2 – Find cos(180°):
      From the cosine graph:
      
      🟢 Answer: -1

1. Example 3 – Sketch one full cycle of sin(x):
   Use the following points:
   • (0°, 0)
   • (90°, 1)
   • (180°, 0)
   • (270°, -1)
   • (360°, 0)

   Connect the points with a smooth wave!

✍🏽 Practice Exercises

🅰️ Section A – Understanding Graphs

1. What is the period of the basic sine graph?
2. What is the amplitude of y = cos(x)?

🅱️ Section B – Value Interpretation

1. Find the value of .
2. Find the value of .
3. Estimate from the graph.

🅾️ Section C – Real-Life Application

1. A ferris wheel follows a sine curve. If the height varies from 0 m to 20 m, what is the amplitude?

✅ Answers and Explanations

• A1: 360° (or radians)
• A2: Amplitude is 1
• B1:
• B2:
• B3:
• C1: Amplitude =

📌 Recap

• 📈 The sine and cosine graphs are periodic and wave-shaped
• 🔄 They repeat every 360° and range from -1 to 1
• ⚡ They’re used to model motion, sound, electricity, and more!

💭 Reflection Prompt

Can you find any wave-like motion around you? Think of swings, music, or light. How could a sine or cosine graph represent that movement?