02-07 - Unit Circle and Radians
Phase: 2 | Subject: 02-07 Prerequisites: 02-05-pythagoras-and-right-triangle-trig.md (SOH CAH TOA) Next subject: 02-08-trigonometric-functions.md
Learning Objectives
By the end of this subject, you will be able to:
- Convert between degrees and radians
- Use the unit circle to find exact trig values
- Find exact values at standard angles (0°, 30°, 45°, 60°, 90°)
- Use reference angles for any quadrant
- Calculate arc length and sector area in radians
Core Content
Degrees vs Radians
Degrees: Based on dividing a circle into 360 parts Radians: Based on arc length relative to radius
$1 radian = angle where arc length = radius $
Conversion
$Radians = Degrees × π/180 Degrees = Radians × 180/π $
Common conversions: - 180° = π radians - 90° = π/2 radians - 60° = π/3 radians - 45° = π/4 radians - 30° = π/6 radians - 360° = 2π radians
The Unit Circle
⚠️ THIS IS CRITICAL — the unit circle unifies all of trigonometry. Understanding that (cos θ, sin θ) are coordinates on a circle of radius 1 is the key insight that makes everything from trig identities to Fourier transforms make sense.
A circle with radius 1 centred at the origin. Any point on it has coordinates (cos θ, sin θ).
Key insight: For any angle θ, if you go θ radians from (1, 0) counter-clockwise: - x-coordinate = cos(θ) - y-coordinate = sin(θ)
Exact Values from Special Triangles
45° (π/4) — from isosceles right triangle (1, 1, √2)
- sin(45°) = √2/2
- cos(45°) = √2/2
- tan(45°) = 1
30° (π/6) and 60° (π/3) — from equilateral triangle split in half (1, √3, 2)
- sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3
- sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3
Memory trick: "SOH CAH TOA" for 30°/60°: - sin(30) = opposite/hypotenuse = 1/2 (smaller over 2) - cos(30) = adjacent/hypotenuse = √3/2 (larger over 2)
Reference Angles
For any angle in any quadrant, the reference angle is the acute angle it makes with the x-axis.
| Quadrant | Reference angle α |
|---|---|
| I (0 to π/2) | θ |
| II (π/2 to π) | π - θ |
| III (π to 3π/2) | θ - π |
| IV (3π/2 to 2π) | 2π - θ |
Sign rules (ASTC): - All positive in Quadrant I - Sin positive in Quadrant II - Tan positive in Quadrant III - Cos positive in Quadrant IV
Arc Length and Sector Area in Radians
Arc Length
$s = rθ $
where θ is in RADIANS
Sector Area
$A = (1/2)r²θ $
where θ is in RADIANS
Why this is simpler: The 2π in the degree formulas cancels nicely when using radians.
Key Terms
- 02 07 Unit Circle And Radians
- 45° (π/4) — from isosceles right triangle (1, 1, √2)
- Arc Length
- Arc Length and Sector Area in Radians
- Conversion
- Correct: A)
- Correct: B)
- Correct: D)
- Degrees vs Radians
- Exact Values from Special Triangles
- Example 1: Convert 120° to radians
- Example 2: Convert 5π/6 radians to degrees
Worked Examples
Example 1: Convert 120° to radians
120° × π/180 = 2π/3 radians
Example 2: Convert 5π/6 radians to degrees
(5π/6) × 180/π = 150°
Example 3: Find sin(300°)
300° is in Quadrant IV (cos positive, sin negative). Reference angle: 360° - 300° = 60° sin(300°) = -sin(60°) = -√3/2
Example 4: Arc length
Circle radius 5, angle 2π/3 radians. s = 5 × (2π/3) = 10π/3
Quiz
Q1: What does the concept of Arc Length primarily refer to in this subject?
A) A computational error related to Arc Length B) A visual representation of Arc Length C) A historical anecdote about Arc Length D) The definition and application of Arc Length
Correct: D)
- If you chose A: This is incorrect. Arc Length is defined as: the definition and application of arc length. The other options describe different aspects that are not the primary focus.
- If you chose B: This is incorrect. Arc Length is defined as: the definition and application of arc length. The other options describe different aspects that are not the primary focus.
- If you chose C: This is incorrect. Arc Length is defined as: the definition and application of arc length. The other options describe different aspects that are not the primary focus.
- If you chose D: Arc Length is defined as: the definition and application of arc length. The other options describe different aspects that are not the primary focus. Correct!
Q2: What is the primary purpose of Arc Length and Sector Area in Radians?
A) It is used to arc length and sector area in radians in mathematical analysis B) It is used only in advanced research contexts C) It is primarily a historical notation system D) It replaces all other methods in this domain
Correct: A)
- If you chose A: Arc Length and Sector Area in Radians serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
- If you chose B: This is incorrect. Arc Length and Sector Area in Radians serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose C: This is incorrect. Arc Length and Sector Area in Radians serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose D: This is incorrect. Arc Length and Sector Area in Radians serves the purpose described in the correct answer. The other options misrepresent its role.
Q3: Which statement about Conversion is TRUE?
A) Conversion is an advanced topic beyond this subject's scope B) Conversion is mentioned only as a historical footnote C) Conversion is not related to this subject D) Conversion is a fundamental concept covered in this subject
Correct: D)
- If you chose A: This is incorrect. Conversion is a fundamental concept covered in this subject. This subject covers Conversion as part of its core content.
- If you chose B: This is incorrect. Conversion is a fundamental concept covered in this subject. This subject covers Conversion as part of its core content.
- If you chose C: This is incorrect. Conversion is a fundamental concept covered in this subject. This subject covers Conversion as part of its core content.
- If you chose D: Conversion is a fundamental concept covered in this subject. This subject covers Conversion as part of its core content. Correct!
Q4: Based on the worked examples in this subject, what is the correct result?
A) 7π/6 B) An unrelated numerical value C) A different result from a common mistake D) The inverse of the correct answer
Correct: A)
- If you chose A: The worked examples show that the result is 7π/6. The other options represent common errors. Correct!
- If you chose B: This is incorrect. The worked examples show that the result is 7π/6. The other options represent common errors.
- If you chose C: This is incorrect. The worked examples show that the result is 7π/6. The other options represent common errors.
- If you chose D: This is incorrect. The worked examples show that the result is 7π/6. The other options represent common errors.
Q5: How are Conversion and Degrees vs Radians related?
A) Conversion and Degrees vs Radians are completely unrelated topics B) Conversion and Degrees vs Radians are closely related concepts C) Conversion is a special case of Degrees vs Radians D) Conversion is the inverse of Degrees vs Radians
Correct: B)
- If you chose A: This is incorrect. Both Conversion and Degrees vs Radians are covered in this subject as interconnected topics.
- If you chose B: Both Conversion and Degrees vs Radians are covered in this subject as interconnected topics. Correct!
- If you chose C: This is incorrect. Both Conversion and Degrees vs Radians are covered in this subject as interconnected topics.
- If you chose D: This is incorrect. Both Conversion and Degrees vs Radians are covered in this subject as interconnected topics.
Q6: What is a common pitfall when working with Exact Values from Special Triangles?
A) The main error with Exact Values from Special Triangles is using it when it is not needed B) Exact Values from Special Triangles is always computed the same way in all contexts C) Exact Values from Special Triangles has no common misconceptions D) A common mistake is confusing Exact Values from Special Triangles with a similar concept
Correct: D)
- If you chose A: This is incorrect. Students often confuse Exact Values from Special Triangles with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose B: This is incorrect. Students often confuse Exact Values from Special Triangles with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose C: This is incorrect. Students often confuse Exact Values from Special Triangles with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose D: Students often confuse Exact Values from Special Triangles with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
Q7: When should you apply The Unit Circle?
A) Use The Unit Circle only in pure mathematics contexts B) Avoid The Unit Circle unless explicitly instructed C) The Unit Circle is not practically useful D) Apply The Unit Circle to solve problems in this subject's domain
Correct: D)
- If you chose A: This is incorrect. The Unit Circle is a practical tool used throughout this subject to solve relevant problems.
- If you chose B: This is incorrect. The Unit Circle is a practical tool used throughout this subject to solve relevant problems.
- If you chose C: This is incorrect. The Unit Circle is a practical tool used throughout this subject to solve relevant problems.
- If you chose D: The Unit Circle is a practical tool used throughout this subject to solve relevant problems. Correct!
Practice Problems
-
Convert 210° to radians Answer: 210 × π/180 = 7π/6
-
Convert 3π/4 radians to degrees Answer: (3π/4) × 180/π = 135°
-
Exact value: sin(π/3) Answer: √3/2
-
Exact value: cos(5π/6) Answer: -√3/2 (Quadrant II, reference angle π/6)
-
Exact value: tan(3π/4) Answer: -1 (Quadrant II, tan negative, tan(π/4) = 1)
-
Arc length with r = 4, θ = π/3 Answer: 4 × π/3 = 4π/3
-
Sector area with r = 6, θ = π/2 Answer: (1/2) × 36 × π/2 = 9π
Summary
Key takeaways:
- π radians = 180°
- Unit circle: point at angle θ is (cos θ, sin θ)
- Exact values from 30-60-90 and 45-45-90 triangles
- Reference angles reduce any angle to Quadrant I
- ASTC: All/Sin/Tan/Cos positive by quadrant
- Arc length: s = rθ (θ in radians)
- Sector area: A = (1/2)r²θ (θ in radians)
Pitfalls
- Using degrees in arc length and sector area: The formulas s = rθ and A = (1/2)r²θ require θ in RADIANS. If your angle is in degrees, convert first (multiply by π/180). Plugging degrees directly into these formulas gives nonsense results.
- Mixing up sin and cos values at special angles: sin(30°) = 1/2 and cos(30°) = √3/2. Many students swap them. Remember: for 30°, the smaller coordinate (1/2) is the sine; for 60°, the smaller coordinate (1/2) is the cosine.
- Getting quadrant signs wrong (ASTC): After finding the reference angle, you must apply the correct sign for the quadrant. Quadrant II: only sine positive. Quadrant III: only tan positive. Quadrant IV: only cos positive. Forgetting this sign adjustment is one of the most frequent errors.
- Incorrect reference angle calculation: The reference angle is always measured from the terminal side to the x-axis, not the y-axis. In Quadrant II: π - θ; Quadrant III: θ - π; Quadrant IV: 2π - θ.
- Confusing radians and degrees in the unit circle: π/6 = 30°, not 60°. π/3 = 60°, not 30°. These two are frequently swapped. Memorise: π/6 = 30° (smaller fraction, smaller angle); π/3 = 60°.
Next Steps
Next up: 02-08-trigonometric-functions.md