Lesson 2: Trig Ratios of 30°, 45°, and 60°

👨🏽‍🏫 Introduction

Ever noticed how some angles keep popping up in design, construction, and nature? ⛺📐 The angles 30°, 45°, and 60° are special because they come from simple triangles and give us exact trigonometric values—no calculator needed!

In this lesson, you’ll learn how to memorize and use the sine, cosine, and tangent values for these special angles. These values are essential for solving right triangles faster, especially in exams and real-life situations.

• 🔢 Recall sin, cos, and tan values for 30°, 45°, and 60°
• 📐 Understand where these values come from using special triangles
• 🛠 Apply them to solve triangle problems more efficiently

🧠 Key Concepts

📌 1. Special Triangles

45°–45°–90° triangle: An isosceles right triangle. The two legs are equal, and the hypotenuse is times a leg.

30°–60°–90° triangle: Comes from cutting an equilateral triangle in half. The sides are in a ratio of .

📌 2. Exact Trig Values

Angle (°)	sin(θ)	cos(θ)	tan(θ)
30°
45°
60°

🔍 Worked Examples

1. 1. Example 1 – Use sin(30°) to find a side:
      If the hypotenuse is 10 cm, what is the side opposite 30°?
      
      🟢 Answer: 5 cm

1. 1. Example 2 – Use cos(60°) to find a side:
      Hypotenuse = 8 cm. Find adjacent side.
      
      🟢 Answer: 4 cm

1. Example 3 – Use tan(45°) to find a missing side:
   If opposite side = 7 cm, and , find the adjacent side.
   
   🟢 Answer: 7 cm

✍🏽 Practice Exercises

🅰️ Section A – Recall Values

1. Write down the exact values of sin(60°), cos(30°), and tan(45°).
2. Which of the three angles (30°, 45°, 60°) has equal sin and cos values?

🅱️ Section B – Solve Using Special Ratios

1. Find the side opposite 60° in a triangle with hypotenuse = 14 cm.
2. If tan(30°) = 0.577 and opposite = 5 cm, find the adjacent side.
3. In a 45° right triangle with one leg = 6 cm, find the hypotenuse.

✅ Answers and Explanations

• A1: sin(60°) = , cos(30°) = , tan(45°) = 1
• A2: 45° – because sin(45°) = cos(45°)
• B1:
• B2:
• B3: Hypotenuse =

📌 Recap

• 📐 Special right triangles (30°–60°–90° and 45°–45°–90°) give us exact trig values
• 📊 Memorizing these helps solve many problems quickly, without a calculator
• 🎯 Use them to calculate unknown sides or angles in triangle problems

💭 Reflection Prompt

Try to sketch a right triangle and label it with one of the special angles (30°, 45°, or 60°). Can you calculate all the side lengths using the known trig values?