Lesson 1: Sine, Cosine, and Tangent of Angles

👨🏽‍🏫 Introduction

Have you ever tried to measure the height of a tall tree without climbing it? 🌳 Or figure out the slope of a hill? 🏞️ Trigonometry makes all of this possible!

This lesson introduces the 3 most important trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These tools help us connect the sides of a right-angled triangle with its angles—and they appear in everything from architecture and astronomy to navigation and engineering.

• 📐 Define and understand sine, cosine, and tangent
• 🧠 Learn the SOH-CAH-TOA memory trick
• 📏 Apply trigonometry to solve real problems involving right-angled triangles

🧠 Key Concepts

📌 1. Right-Angled Triangles

In a right-angled triangle:

• The hypotenuse is the longest side (opposite the right angle)
• The opposite side is across from the angle you’re measuring
• The adjacent side is next to the angle you’re measuring (not the hypotenuse)

📌 2. Trigonometric Ratios

These are the three main trig ratios used in right-angled triangles:

• Sine:
• Cosine:
• Tangent:

🎯 SOH-CAH-TOA helps you remember:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

🔍 Worked Examples

1. 1. Example 1 – Find sin(θ):
      A triangle has an opposite side of 3 cm and hypotenuse of 5 cm. Find .
      
      🟢 Answer: 0.6

1. 1. Example 2 – Find cos(θ):
      The adjacent side is 4 cm, and the hypotenuse is 5 cm.
      
      🟢 Answer: 0.8

1. 1. Example 3 – Find tan(θ):
      The opposite side is 3 cm, and the adjacent side is 4 cm.
      
      🟢 Answer: 0.75

1. Example 4 – Find the missing side using sine:
   If and the hypotenuse is 10 cm, what is the opposite side?
   
   🟢 Answer: 5 cm

✍🏽 Practice Exercises

🅰️ Section A – Identify Ratios

1. Define sine, cosine, and tangent in terms of triangle sides.
2. Which side is the hypotenuse in a right-angled triangle?

🅱️ Section B – Calculate Ratios

1. In a right triangle, opposite = 6 cm, hypotenuse = 10 cm. Find sin(θ).
2. Adjacent = 5 cm, hypotenuse = 13 cm. Find cos(θ).
3. Opposite = 8 cm, adjacent = 6 cm. Find tan(θ).

🅾️ Section C – Real-Life Application

1. A ladder leans against a wall making a 60° angle with the ground. If the ladder is 12 m long, how high up the wall does it reach?

✅ Answers and Explanations

• A1:
  sin(θ) = Opposite/Hypotenuse,
  cos(θ) = Adjacent/Hypotenuse,
  tan(θ) = Opposite/Adjacent
• A2: The hypotenuse is the side opposite the 90° angle (longest side)
• B1:
• B2:
• B3:
• C1:
  🟢 So the height is approximately 10.39 m

📌 Recap

• 📐 Trigonometry deals with the relationships between angles and sides in right-angled triangles
• 🧠 Remember SOH-CAH-TOA for sine, cosine, and tangent
• 🎯 You can solve for missing angles or sides using these ratios

💭 Reflection Prompt

Next time you see a ladder, a roof, or even a slide, try to estimate the angle it makes with the ground. Could you use trigonometry to calculate its height or length?