S3 Mathematics - Term 1

Topic 1: Trigonometry I

Lesson 3: Sine, Cosine, and Tangent Ratios

Duration: 40 minutes

Learning Outcomes

By the end of this lesson, you should be able to:

• Define sine, cosine, and tangent ratios
• Apply SOH-CAH-TOA to identify the correct ratio
• Calculate trig ratios from triangle measurements

Review

From the unit circle:

• Each angle gives us x and y coordinates
• These represent sides of a right triangle
• Today: we give these ratios special names!

The Three Trigonometric Ratios

For a right-angled triangle with angle θ:

1. Sine (sin θ)
2. Cosine (cos θ)
3. Tangent (tan θ)

Sine

sin θ = opposite ÷ hypotenuse

Example:

• Opposite = 3 cm
• Hypotenuse = 5 cm
• sin θ = 3/5 = 0.6

Cosine

cos θ = adjacent ÷ hypotenuse

Example:

• Adjacent = 4 cm
• Hypotenuse = 5 cm
• cos θ = 4/5 = 0.8

Tangent

tan θ = opposite ÷ adjacent

Example:

• Opposite = 3 cm
• Adjacent = 4 cm
• tan θ = 3/4 = 0.75

All Three Together

For the same triangle:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent

All three are ratios that depend on angle θ

SOH-CAH-TOA

Let's Chant!

SOH-CAH-TOA
SOH-CAH-TOA
SOH-CAH-TOA

Say it together three times!

Memory Phrase

Some Old Hens
Cackle And Howl
Till Old Age

Create your own phrase to remember!

Worked Example

Pair Practice 1

Triangle with angle B:

• Opposite = 5 cm
• Hypotenuse = 13 cm

1. Find the adjacent side (use Pythagoras)
2. Calculate sin B, cos B, tan B

Answer 1

Adjacent = √(13² - 5²) = √144 = 12 cm

• sin B = 5/13 ≈ 0.385
• cos B = 12/13 ≈ 0.923
• tan B = 5/12 ≈ 0.417

Pair Practice 2

Triangle with angle C:

• Adjacent = 7 cm
• Opposite = 24 cm

1. Find the hypotenuse
2. Calculate sin C, cos C, tan C

Answer 2

Hypotenuse = √(7² + 24²) = √625 = 25 cm

• sin C = 24/25 = 0.96
• cos C = 7/25 = 0.28
• tan C = 24/7 ≈ 3.43

Key Points

✓ All three ratios are fractions (divisions)
✓ They compare two sides of a triangle
✓ The angle determines the ratio values
✓ SOH-CAH-TOA helps you remember!

Summary

Exit Questions

Triangle with angle θ:

• Opposite = 8 cm
• Adjacent = 6 cm
• Hypotenuse = 10 cm

Find: sin θ, cos θ, tan θ

Answers

• sin θ = 8/10 = 0.8
• cos θ = 6/10 = 0.6
• tan θ = 8/6 ≈ 1.33

Homework

1. Write the three trig ratio definitions
2. Triangle: opposite = 12, hyp = 13
   Find adjacent, then calculate all ratios
3. Triangle: adjacent = 15, opposite = 8
   Find hypotenuse, then calculate all ratios
4. Create your own SOH-CAH-TOA phrase!

Next Lesson

Lesson 4: Using Calculators for Trig Values

• Find sin 37° without measuring!
• Learn to use scientific calculators
• Discover trig values for any angle

Credits

Created: December 2025
Based on: NCDC Lower Secondary Mathematics Syllabus (2019)

Source: National Curriculum Development Centre (NCDC), Uganda