📐 Concept diagram

02-05 - Pythagoras and Right Triangle Trig

Phase: 2 | Subject: 02-05 Prerequisites: 02-04-perimeter-area-volume.md (area of triangles), 01-04-coordinate-geometry-2d.md (distance formula) Next subject: 02-06-non-right-triangle-trig.md

Learning Objectives

By the end of this subject, you will be able to:

1. State and apply Pythagoras' theorem
2. Use Pythagorean triples for quick calculations
3. Apply 3D Pythagoras in three dimensions
4. Use sine, cosine, and tangent ratios (SOH CAH TOA)
5. Solve for missing sides and angles in right triangles
6. Apply trigonometry to angles of elevation and depression
7. Understand and calculate bearings

Core Content

Pythagoras' Theorem

⚠️ THIS IS CRITICAL — a² + b² = c² is arguably the most important formula in all of geometry. It appears everywhere: from distance calculations to the unit circle to vector magnitudes to machine learning loss functions.

In any RIGHT-angled triangle:

$a² + b² = c²
$

where c = hypotenuse (longest side, opposite the right angle) and a, b = the two shorter sides (legs)

Why it works: The area of the square on the hypotenuse equals the sum of areas of squares on the other two sides. This can be proven geometrically.

Finding the hypotenuse: $c = √(a² + b²)$

Finding a shorter side: $a = √(c² - b²)$ or $b = √(c² - a²)$

Pythagorean Triples

Integer solutions to a² + b² = c²:

Multiples work too: (6, 8, 10), (9, 12, 15), etc.

3D Pythagoras

For a rectangular box with dimensions a, b, c:

Diagonal from one corner to opposite: d = √(a² + b² + c²)

Example: Box 3m × 4m × 5m d = √(9 + 16 + 25) = √50 = 5√2 ≈ 7.07m

SOH CAH TOA Mnemonic

For a RIGHT triangle with angle θ:

⚠️ THIS IS CRITICAL — SOH CAH TOA is the single most important mnemonic in trigonometry. Memorise it. You will use it in every trig subject from now through calculus and beyond.

$sin(θ) = opposite / hypotenuse  (SOH)
cos(θ) = adjacent / hypotenuse  (CAH)
tan(θ) = opposite / adjacent     (TOA)
$

To remember: - SOH: Sine = Opposite over Hypotenuse - CAH: Cosine = Adjacent over Hypotenuse - TOA: Tangent = Opposite over Adjacent

Finding Missing Sides

If you know an angle and one side, use the trig ratios.

Example: Right triangle with θ = 30°, hypotenuse = 10. Find opposite side.

sin(30°) = opposite / 10 opposite = 10 × sin(30°) = 10 × 0.5 = 5

Finding Missing Angles

Use inverse trig functions: sin⁻¹, cos⁻¹, tan⁻¹

Example: Opposite = 4, adjacent = 3. Find θ.

tan(θ) = 4/3 θ = tan⁻¹(4/3) ≈ 53.13°

When to Use Which Ratio

Pitfalls

• SOH CAH TOA order matters: Many beginners swap opposite and adjacent. Draw the triangle and label the sides relative to the angle you're working with BEFORE picking a ratio.
• Using the wrong inverse: If you want an angle from opposite and hypotenuse, use sin⁻¹. From opposite and adjacent, use tan⁻¹. Using the wrong one gives a wrong angle.
• Pythagoras only works for RIGHT triangles. If the triangle isn't right-angled, use sine/cosine rules (02-06).
• Bearings are clockwise from North. A common mistake is measuring from East or going anticlockwise.

Angles of Elevation and Depression

Angle of elevation: Looking UP from horizontal to an object Angle of depression: Looking DOWN from horizontal to an object

By parallel line rules, angle of elevation = angle of depression.

Example: Standing 50m from a building. Angle of elevation to top is 35°. Find building height.

tan(35°) = height / 50 height = 50 × tan(35°) ≈ 50 × 0.700 ≈ 35.0m

Bearings

A bearing is a direction measured clockwise from North.

• Always written as three digits: 000° to 360°
• North = 000° or 360°
• East = 090°
• South = 180°
• West = 270°

Example: Bearing of 135° is Southeast (halfway between South and East)

Back bearing: Add or subtract 180° (if result > 360°, subtract 360°)

Example: Bearing 045° → back bearing 225° Bearing 300° → back bearing 120° (300 - 180 = 120)

Key Terms

• 02 05 Pythagoras And Right Triangle Trig
• 3D Pythagoras
• Angles of Elevation and Depression
• Bearings
• Bearings are clockwise from North.
• Common Pitfalls
• Correct: A)
• Correct: B)
• Correct: C)
• Example 1: Pythagoras
• Example 2: Trigonometry
• Example 3: Angle of depression

Worked Examples

Example 1: Pythagoras

Right triangle with legs 6cm and 8cm. Find hypotenuse.

c = √(36 + 64) = √100 = 10cm

Recognise this as the (6, 8, 10) triple (multiple of 3-4-5).

Example 2: Trigonometry

Right triangle: angle 40°, adjacent side 12cm. Find opposite.

tan(40°) = opposite / 12 opposite = 12 × tan(40°) ≈ 12 × 0.839 ≈ 10.07cm

Example 3: Angle of depression

From a cliff 80m high, angle of depression to a boat is 25°. Find horizontal distance to boat.

The angle of depression = angle of elevation from boat = 25°. tan(25°) = 80 / distance distance = 80 / tan(25°) ≈ 80 / 0.466 ≈ 171.7m

Quiz

Q1: What does the concept of Angles of Elevation and Depression primarily refer to in this subject?

A) The definition and application of Angles of Elevation and Depression B) A historical anecdote about Angles of Elevation and Depression C) A visual representation of Angles of Elevation and Depression D) A computational error related to Angles of Elevation and Depression

Correct: A)

• If you chose A: Angles of Elevation and Depression is defined as: the definition and application of angles of elevation and depression. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Angles of Elevation and Depression is defined as: the definition and application of angles of elevation and depression. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Angles of Elevation and Depression is defined as: the definition and application of angles of elevation and depression. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Angles of Elevation and Depression is defined as: the definition and application of angles of elevation and depression. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Bearings?

A) It is primarily a historical notation system B) It replaces all other methods in this domain C) It is used only in advanced research contexts D) It is used to bearings in mathematical analysis

Correct: D)

• If you chose A: This is incorrect. Bearings serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Bearings serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Bearings serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: Bearings serves the purpose described in the correct answer. The other options misrepresent its role. Correct!

Q3: Which statement about Common Pitfalls is TRUE?

A) Common Pitfalls is not related to this subject B) Common Pitfalls is a fundamental concept covered in this subject C) Common Pitfalls is mentioned only as a historical footnote D) Common Pitfalls is an advanced topic beyond this subject's scope

Correct: B)

• If you chose A: This is incorrect. Common Pitfalls is a fundamental concept covered in this subject. This subject covers Common Pitfalls as part of its core content.
• If you chose B: Common Pitfalls is a fundamental concept covered in this subject. This subject covers Common Pitfalls as part of its core content. Correct!
• If you chose C: This is incorrect. Common Pitfalls is a fundamental concept covered in this subject. This subject covers Common Pitfalls as part of its core content.
• If you chose D: This is incorrect. Common Pitfalls is a fundamental concept covered in this subject. This subject covers Common Pitfalls as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) An unrelated numerical value C) ** Bearing 045° → back bearing 225° D) A different result from a common mistake

Correct: C)

• If you chose A: This is incorrect. The worked examples show that the result is ** Bearing 045° → back bearing 225°. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is ** Bearing 045° → back bearing 225°. The other options represent common errors.
• If you chose C: The worked examples show that the result is ** Bearing 045° → back bearing 225°. The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is ** Bearing 045° → back bearing 225°. The other options represent common errors.

Q5: How are Common Pitfalls and Pythagoras' Theorem related?

A) Common Pitfalls and Pythagoras' Theorem are completely unrelated topics B) Common Pitfalls is a special case of Pythagoras' Theorem C) Common Pitfalls and Pythagoras' Theorem are closely related concepts D) Common Pitfalls is the inverse of Pythagoras' Theorem

Correct: C)

• If you chose A: This is incorrect. Both Common Pitfalls and Pythagoras' Theorem are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Common Pitfalls and Pythagoras' Theorem are covered in this subject as interconnected topics.
• If you chose C: Both Common Pitfalls and Pythagoras' Theorem are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both Common Pitfalls and Pythagoras' Theorem are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Pythagorean Triples?

A) The main error with Pythagorean Triples is using it when it is not needed B) A common mistake is confusing Pythagorean Triples with a similar concept C) Pythagorean Triples is always computed the same way in all contexts D) Pythagorean Triples has no common misconceptions

Correct: B)

• If you chose A: This is incorrect. Students often confuse Pythagorean Triples with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse Pythagorean Triples with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse Pythagorean Triples with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Pythagorean Triples with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply 3D Pythagoras?

A) Use 3D Pythagoras only in pure mathematics contexts B) 3D Pythagoras is not practically useful C) Apply 3D Pythagoras to solve problems in this subject's domain D) Avoid 3D Pythagoras unless explicitly instructed

Correct: C)

• If you chose A: This is incorrect. 3D Pythagoras is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. 3D Pythagoras is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: 3D Pythagoras is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. 3D Pythagoras is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Find hypotenuse of right triangle with legs 5 and 12 Answer: √(25 + 144) = √169 = 13

2. Find missing side if hypotenuse = 10 and one leg = 6 Answer: √(100 - 36) = √64 = 8

3. 3D: diagonal of 3×4×12 box Answer: √(9 + 16 + 144) = √169 = 13

4. Right triangle: angle 45°, hypotenuse 10. Find opposite. Answer: 10 × sin(45°) = 10 × (√2/2) = 5√2 ≈ 7.07

5. Right triangle: opposite 7, adjacent 24. Find angle. Answer: tan⁻¹(7/24) ≈ 16.26°

6. Angle of elevation to top of tree (height 15m) from 20m away Answer: tan⁻¹(15/20) = tan⁻¹(0.75) ≈ 36.87°

7. Bearing 210°. What direction is this? Answer: 210 - 180 = 30° past South, so South 30° West (or just Southwest-ish)

Summary

Key takeaways:

• Pythagoras: a² + b² = c² (only for right triangles)
• 3D Pythagoras: d² = a² + b² + c²
• SOH CAH TOA for right triangle trig ratios
• Use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) to find angles
• Angle of elevation = angle of depression (alternate interior angles)
• Bearings: clockwise from North, always 3 digits

Next Steps

Next up: 02-06-non-right-triangle-trig.md