S3 Mathematics - Term 1

Topic 1: Trigonometry I

Lesson 6: Solving Right Triangles

Duration: 40 minutes

Learning Outcomes

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

• Solve for all missing parts of a right-angled triangle
• Choose appropriate trig ratios for given information
• Apply both forward and inverse trig functions systematically

What Does "Solve" Mean?

Solving a triangle = Find ALL missing sides and angles

Use:

• The given information
• Systematic approach
• All our trig tools!

Tools Available

1. SOH-CAH-TOA (find sides from angles)
2. sin⁻¹, cos⁻¹, tan⁻¹ (find angles from sides)
3. Pythagoras (a² + b² = c²)
4. Angle sum (angles add to 180 degrees)

What Might We Be Given?

• One angle + one side
• Two sides

(Plus we always have the 90 degrees angle!)

Solving Strategy

Example 1: Angle + Hypotenuse

Given: angle A = 35 degrees, hypotenuse = 20 cm

Find: all other parts

Solution to Example 1

Angle B: 180 - 90 - 35 = 55 degrees

Opposite: sin 35 degrees = opp/20
opp = 20 × 0.574 = 11.47 cm

Adjacent: cos 35 degrees = adj/20
adj = 20 × 0.819 = 16.38 cm

Check: 11.47² + 16.38² ≈ 20² ✓

Example 2: Angle + Opposite

Given: angle theta = 40 degrees, opposite = 12 cm

Find: all other parts

Solution to Example 2

Angle: 180 - 90 - 40 = 50 degrees

Hypotenuse: sin 40 degrees = 12/hyp
hyp = 12/0.643 ≈ 18.67 cm

Adjacent: tan 40 degrees = 12/adj
adj = 12/0.839 ≈ 14.30 cm

Pair Practice

Given: angle = 55 degrees, adjacent = 10 cm

Find: all other parts

Answer

• Other angle = 35 degrees
• Opposite: tan 55 degrees = opp/10
  opp ≈ 14.28 cm
• Hypotenuse: cos 55 degrees = 10/hyp
  hyp ≈ 17.43 cm

Example 3: Two Sides Given

Given: hypotenuse = 25 cm, opposite = 15 cm

Find: all other parts

Solution to Example 3

Adjacent: √(25² - 15²) = √400 = 20 cm

Angle A: sin A = 15/25 = 0.6
A = sin⁻¹(0.6) ≈ 36.9 degrees

Angle B: 180 - 90 - 36.9 = 53.1 degrees

Complete Example

Key Strategy Reminder

Given angle + side:

1. Find third angle (180 degrees)
2. Use sin/cos/tan for sides

Given two sides:

1. Find third side (Pythagoras)
2. Use sin⁻¹/cos⁻¹/tan⁻¹ for angles

Group Practice 1

Given: angle = 28 degrees, hypotenuse = 30 cm

Find: all parts

Answer 1

• Other angle = 62 degrees
• Opposite = 30 × sin 28 degrees ≈ 14.08 cm
• Adjacent = 30 × cos 28 degrees ≈ 26.49 cm

Group Practice 2

Given: adjacent = 9 cm, hypotenuse = 15 cm

Find: all parts

Answer 2

• Opposite = √(15² - 9²) = 12 cm
  (This is a 3-4-5 triangle scaled by 3!)
• Angle = cos⁻¹(9/15) ≈ 53.1 degrees
• Other angle ≈ 36.9 degrees

Real-World Application

Problem: A ladder 5 m long leans against a wall. The base is 2 m from the wall.

1. How high up the wall does it reach?
2. What angle with the ground?

Solution

Height: √(5² - 2²) = √21 ≈ 4.58 m

Angle: cos theta = 2/5
theta = cos⁻¹(0.4) ≈ 66.4 degrees

Checking Your Work

Three ways to verify:

1. Pythagoras: a² + b² = c²
2. Angle sum: angles = 180 degrees
3. Forward check: Use found angle to recalculate a side

Always check!

Problem-Solving Tips

1. Draw and label the triangle
2. Write down what you know
3. Decide what to find first
4. Use appropriate ratio
5. Check your answer!

Exit Problem

Given: angle A = 32 degrees, opposite = 8 cm

Find: all other sides and angles

Answer

• Angle B = 58 degrees
• Hypotenuse = 8/sin 32 degrees ≈ 15.09 cm
• Adjacent = 8/tan 32 degrees ≈ 12.80 cm

Check: 8² + 12.80² ≈ 15.09² ✓

Homework

Solve completely (find all missing parts):

1. Angle = 45 degrees, opposite = 10 cm
2. Angle = 60 degrees, hypotenuse = 20 cm
3. Opposite = 5 cm, adjacent = 12 cm
4. Adjacent = 9 cm, hypotenuse = 15 cm
5. Word problem: Kite string

Next Lesson

Lesson 7: Angles of Elevation and Depression

• Special applications of triangle solving
• Real-world problems
• Looking up and down!

Summary

You now have ALL the tools to solve any right triangle!

• Forward trig (sin, cos, tan)
• Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹)
• Pythagoras
• Angle sum

Practice makes perfect!

Credits

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

Source: National Curriculum Development Centre (NCDC), Uganda