📐 Concept diagram

02-06 - Non-Right Triangle Trig

Phase: 2 | Subject: 02-06 Prerequisites: 02-05-pythagoras-and-right-triangle-trig.md Next subject: 02-07-unit-circle-and-radians.md

Learning Objectives

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

1. Apply the sine rule to find unknown sides and angles
2. Apply the cosine rule to find unknown sides and angles
3. Calculate triangle area using (1/2)ab·sin(C)
4. Recognise and resolve the ambiguous case (SSA)
5. Apply trigonometry to navigation and bearing problems

Core Content

Sine Rule

⚠️ THIS IS CRITICAL — the sine and cosine rules are your toolkit for ANY triangle, not just right triangles. Most real-world trig problems involve non-right triangles.

For ANY triangle ABC:

$a / sin(A) = b / sin(B) = c / sin(C)
$

where a is the side opposite angle A, b opposite B, c opposite C.

Use when you know: - Two angles and one side (AAS or ASA) → find a side - Two sides and a non-included angle (SSA) → find an angle (ambiguous case!)

Finding a Side

Example: In triangle ABC, A = 30°, B = 45°, a = 10. Find b.

a / sin(A) = b / sin(B) 10 / sin(30°) = b / sin(45°) 10 / 0.5 = b / 0.707 20 = b / 0.707 b = 20 × 0.707 ≈ 14.14

Finding an Angle

Example: In triangle ABC, a = 8, b = 10, A = 30°. Find B.

8 / sin(30°) = 10 / sin(B) 8 / 0.5 = 10 / sin(B) 16 = 10 / sin(B) sin(B) = 10/16 = 0.625 B = sin⁻¹(0.625) ≈ 38.68°

Cosine Rule

For ANY triangle ABC:

$a² = b² + c² - 2bc·cos(A)
b² = a² + c² - 2ac·cos(B)
c² = a² + b² - 2ab·cos(C)
$

Use when you know: - Three sides (SSS) → find an angle - Two sides and the INCLUDED angle (SAS) → find the third side

Example: Triangle with a = 7, b = 8, C = 60°. Find c.

c² = 7² + 8² - 2(7)(8)cos(60°) c² = 49 + 64 - 112(0.5) c² = 113 - 56 c² = 57 c = √57 ≈ 7.55

Area Formula

$Area = (1/2)ab·sin(C)
$

where C is the INCLUDED angle between sides a and b.

Example: Two sides 5 and 7 with included angle 30°.

Area = (1/2)(5)(7)sin(30°) = (1/2)(35)(0.5) = 8.75

The Ambiguous Case (SSA)

When you know two sides and a non-included angle, there can be:

• Two triangles (two possible values for the unknown angle)
• One triangle (one solution)
• No triangle (impossible configuration)

Example: a = 10, b = 6, A = 30°

sin(B) = (b·sin(A))/a = (6 × 0.5)/10 = 0.3 B = sin⁻¹(0.3) ≈ 17.46° or 180° - 17.46° = 162.54°

Both are possible if a > b·sin(A) (two triangles) Only one if a = b·sin(A) or a ≥ b (one triangle) None if a < b·sin(A)

Key Terms

• No triangle
• One triangle
• Two triangles

Worked Examples

Example 1: Sine rule — find side

Triangle ABC: A = 40°, B = 65°, a = 12. Find c.

1. Find C: C = 180° - 40° - 65° = 75°
2. Use sine rule: 12 / sin(40°) = c / sin(75°)
3. 12 / 0.643 = c / 0.966
4. 18.67 = c / 0.966
5. c ≈ 18.02

Example 2: Cosine rule — find angle

Triangle with sides 5, 6, 7. Find the largest angle.

1. Largest angle is opposite longest side (7). Let that be C.
2. c² = a² + b² - 2ab·cos(C)
3. 49 = 25 + 36 - 60·cos(C)
4. 49 = 61 - 60·cos(C)
5. 60·cos(C) = 12
6. cos(C) = 0.2
7. C = cos⁻¹(0.2) ≈ 78.46°

Example 3: Area

Triangle with sides 8 and 10, included angle 25°.

Area = (1/2)(8)(10)sin(25°) = 40 × 0.423 ≈ 16.9

Quiz

Q1: What does the concept of Two triangles primarily refer to in this subject?

A) A computational error related to Two triangles B) The definition and application of Two triangles C) A historical anecdote about Two triangles D) A visual representation of Two triangles

Correct: B)

• If you chose A: This is incorrect. Two triangles is defined as: the definition and application of two triangles. The other options describe different aspects that are not the primary focus.
• If you chose B: Two triangles is defined as: the definition and application of two triangles. The other options describe different aspects that are not the primary focus. Correct!
• If you chose C: This is incorrect. Two triangles is defined as: the definition and application of two triangles. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Two triangles is defined as: the definition and application of two triangles. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of One triangle?

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

Correct: C)

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

Q3: Which statement about No triangle is TRUE?

A) No triangle is a fundamental concept covered in this subject B) No triangle is mentioned only as a historical footnote C) No triangle is not related to this subject D) No triangle is an advanced topic beyond this subject's scope

Correct: A)

• If you chose A: No triangle is a fundamental concept covered in this subject. This subject covers No triangle as part of its core content. Correct!
• If you chose B: This is incorrect. No triangle is a fundamental concept covered in this subject. This subject covers No triangle as part of its core content.
• If you chose C: This is incorrect. No triangle is a fundamental concept covered in this subject. This subject covers No triangle as part of its core content.
• If you chose D: This is incorrect. No triangle is a fundamental concept covered in this subject. This subject covers No triangle as part of its core content.

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

A) An unrelated numerical value B) The inverse of the correct answer C) 8 × sin(60°) / sin(50°) ≈ 8 × 0.866 / 0 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 8 × sin(60°) / sin(50°) ≈ 8 × 0.866 / 0. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is 8 × sin(60°) / sin(50°) ≈ 8 × 0.866 / 0. The other options represent common errors.
• If you chose C: The worked examples show that the result is 8 × sin(60°) / sin(50°) ≈ 8 × 0.866 / 0. The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is 8 × sin(60°) / sin(50°) ≈ 8 × 0.866 / 0. The other options represent common errors.

Q5: How are No triangle and Sine Rule related?

A) No triangle is a special case of Sine Rule B) No triangle is the inverse of Sine Rule C) No triangle and Sine Rule are completely unrelated topics D) No triangle and Sine Rule are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both No triangle and Sine Rule are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both No triangle and Sine Rule are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both No triangle and Sine Rule are covered in this subject as interconnected topics.
• If you chose D: Both No triangle and Sine Rule are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Finding A Side?

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

Correct: B)

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

Q7: When should you apply Finding An Angle?

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

Correct: D)

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

Practice Problems

1. Triangle: A = 50°, B = 60°, a = 8. Find b. Answer: C = 70°. 8/sin(50°) = b/sin(60°). b = 8 × sin(60°) / sin(50°) ≈ 8 × 0.866 / 0.766 ≈ 9.05.

2. Triangle: a = 5, b = 7, C = 45°. Find c. Answer: c² = 25 + 49 - 70·cos(45°) = 74 - 49.5 = 24.5. c ≈ 4.95.

3. Triangle sides 3, 4, 5. Find largest angle. Answer: Opposite side 5. cos(C) = (9 + 16 - 25)/12 = 0. C = 90°. Right triangle!

4. Area of triangle with sides 6, 8, included angle 30°. Answer: (1/2)(6)(8)sin(30°) = 24 × 0.5 = 12.

5. SSA case: a = 10, b = 6, A = 40°. How many triangles? Answer: b·sin(A) = 6 × 0.643 = 3.86. Since a = 10 > b = 6, only ONE triangle.

Summary

Key takeaways:

• Sine rule: a/sin(A) = b/sin(B) = c/sin(C) — works for ANY triangle
• Cosine rule: c² = a² + b² - 2ab·cos(C) — works for ANY triangle
• Area formula: (1/2)ab·sin(C)
• Sine rule for AAS/ASA (find side) or SSA (find angle — ambiguous case)
• Cosine rule for SSS (find angle) or SAS (find side)
• Ambiguous case: SSA can give 0, 1, or 2 triangles

Pitfalls

• Using the wrong rule for the situation: Sine rule is for AAS/ASA (find a side) or SSA (find an angle). Cosine rule is for SSS (find an angle) or SAS (find a side). Using sine rule for SSS won't work — you need the cosine rule.
• Forgetting to check the ambiguous case: When given two sides and a non-included angle (SSA), there may be 0, 1, or 2 possible triangles. Always compute h = b·sin(A) to determine how many solutions exist before committing to an answer.
• Mixing up opposite sides and angles: In the sine rule, a is opposite A, b opposite B, c opposite C. Annotating the triangle with its angles and corresponding opposite sides before starting prevents transposition errors.
• Using the wrong included angle in area formula: Area = (1/2)ab·sin(C) requires C to be the angle BETWEEN sides a and b. If you use a non-included angle, the result is wrong.
• Cosine rule sign errors: When solving for an angle with the cosine rule, rearranging c² = a² + b² - 2ab·cos(C) to isolate cos(C) often leads to algebraic slips. Double-check: cos(C) = (a² + b² - c²)/(2ab), not the other way around.

Next Steps

Next up: 02-07-unit-circle-and-radians.md