Mathematics Advanced • Year 11 • Module 2 • Lesson 1

Angles & Radian Measure

Build procedural fluency in converting between degrees and radians, and finding coterminal angles in radian form.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the fundamental identity that anchors every conversion in this lesson:

____________ rad = ____________ °

Q1.2 Fill in the two conversion multipliers:

Degrees → radians: multiply by ________________

Radians → degrees: multiply by ________________

Q1.3 Define a radian in one sentence. (Hint: use the words "arc length" and "radius".)

Stuck? Revisit lesson § What is a radian? and § The two moves.

2. Worked example — converting 135° to radians

Follow each line of algebra. Every step has a reason on the right.

Problem. Convert 135° to radians, leaving the answer in exact form (in terms of π).

Step 1 — Choose the correct conversion direction.

degrees → radians, so multiply by π/180.

Reason: answer should shrink (we're multiplying by < 1).

Step 2 — Substitute the angle and write the product.

135 × π/180 = 135π / 180

Reason: keep π symbolic; never use a decimal approximation until the end.

Step 3 — Cancel the common factor before the final line.

gcd(135, 180) = 45, so 135/180 = 3/4.

∴ 135° = 3π/4 rad.

Reason: HSC markers expect simplified exact form (Trap 01 in the lesson).

Step 4 — Sanity check.

3π/4 ≈ 2.36 rad < π (≈ 3.14). And 135° < 180°. ✓ consistent.

Conclusion. 135° = 3π/4 rad.

3. Faded example — fill in the missing steps

Convert 5π/6 radians to degrees. Fill in each blank line. 4 marks

Step 1 — Choose the direction.

Radians → degrees, so multiply by ________________.

Step 2 — Substitute and cancel π.

(5π/6) × (____ / ____) = 5 × ________ / 6

Step 3 — Compute 180 ÷ 6.

180 ÷ 6 = ________, so the expression becomes 5 × ________ = ________°.

Step 4 — Sanity check.

5π/6 is just less than π, so the answer should be just less than 180°. ________° ____ 180°? (write <, =, or >)

Conclusion. 5π/6 rad = ____________°.

Stuck? Revisit lesson § Worked Example 2 — Radians to Degrees.

4. Graduated practice — convert each angle

For each angle, convert to the other measurement system. Leave radian answers in exact form in terms of π, fully simplified. Show one line of working.

Foundation — the big-six conversions (4 questions)

Q	Given	Working (1 line)	Answer
4.1 1	30° → radians
4.2 1	60° → radians
4.3 1	π/2 rad → degrees
4.4 1	π rad → degrees

Standard — typical HSC difficulty (6 questions)

Show your working — at least one line of substitution and one line of simplification.

4.5 Convert 225° to radians (exact form). 2 marks

4.6 Convert 3π/2 rad to degrees. 2 marks

4.7 Convert −120° to radians (exact form). 2 marks

4.8 Convert 7π/4 rad to degrees. 2 marks

4.9 Convert 540° to radians (exact form). 2 marks

4.10 Find a positive and a negative angle (in radians) coterminal with 7π/4. 2 marks

Extension — combine concepts (2 questions)

4.11 Express 75° in radians as an exact value, then convert that result back to degrees as a check. 3 marks

4.12 Find the smallest positive radian angle that is coterminal with −17π/4. Show every step (you'll need to add multiples of 2π until the result lies in [0, 2π)). 3 marks

Stuck on 4.12? Add 2π = 8π/4 repeatedly until the numerator is non-negative and the magnitude is < 2π.

5. Self-check the easy 3

Tick the first three once you've verified your method works.

For 4.1 (30° → rad) my answer is π/6, and the number is smaller than the original 30 — consistent with multiplying by π/180 < 1.

For 4.3 (π/2 rad → deg) my answer is 90°, and the number is larger than π/2 ≈ 1.57 — consistent with multiplying by 180/π > 1.

For 4.4 (π rad → deg) I got 180°, which matches the anchor identity from the lesson.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Fundamental identity

π rad = 180° (equivalently 2π rad = 360°). Every conversion in the lesson comes from rearranging this.

Q1.2 — Conversion multipliers

Degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π. These are reciprocals.

Q1.3 — Definition of a radian

A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Symbolically, θ (rad) = arc length / radius = l/r.

Q3 — Faded example: 5π/6 rad → degrees

Step 1: multiply by 180/π.
Step 2: (5π/6) × (180/π) = 5 × 180 / 6.
Step 3: 180 ÷ 6 = 30, so 5 × 30 = 150°.
Step 4: 150° < 180°, consistent with 5π/6 < π.
Conclusion: 5π/6 rad = 150°.

Q4.1 — 30° → radians

30 × π/180 = 30π/180 = π/6. (gcd(30, 180) = 30.)

Q4.2 — 60° → radians

60 × π/180 = 60π/180 = π/3.

Q4.3 — π/2 rad → degrees

(π/2) × (180/π) = 180/2 = 90°. (The π cancels.)

Q4.4 — π rad → degrees

π × (180/π) = 180°. (This is the anchor identity.)

Q4.5 — 225° → radians

225 × π/180 = 225π/180. gcd(225, 180) = 45, so 225/180 = 5/4. ∴ 5π/4 rad.

Q4.6 — 3π/2 rad → degrees

(3π/2) × (180/π) = 3 × 180/2 = 3 × 90 = 270°.

Q4.7 — −120° → radians

−120 × π/180 = −120π/180. gcd(120, 180) = 60, so −120/180 = −2/3. ∴ −2π/3 rad. The negative sign carries straight through.

Q4.8 — 7π/4 rad → degrees

(7π/4) × (180/π) = 7 × 180/4 = 7 × 45 = 315°.

Q4.9 — 540° → radians

540 × π/180 = 540π/180. 540/180 = 3 exactly. ∴ 3π rad. (This is one-and-a-half revolutions.)

Q4.10 — Coterminal with 7π/4

Positive coterminal: 7π/4 + 2π = 7π/4 + 8π/4 = 15π/4. Negative coterminal: 7π/4 − 2π = 7π/4 − 8π/4 = −π/4. (Add or subtract 2π, never π — Trap 03 in the lesson.)

Q4.11 — 75° ↔ radians round trip

Forward: 75 × π/180 = 75π/180. gcd(75, 180) = 15, so 75/180 = 5/12. ∴ 75° = 5π/12 rad.
Back-check: (5π/12) × (180/π) = 5 × 180/12 = 5 × 15 = 75°. ✓ Round trip succeeds, so the original conversion is correct.

Q4.12 — Smallest positive coterminal of −17π/4

Add 2π = 8π/4 repeatedly until the result is in [0, 2π):
−17π/4 + 8π/4 = −9π/4. (still negative)
−9π/4 + 8π/4 = −π/4. (still negative)
−π/4 + 8π/4 = 7π/4. (positive and < 2π = 8π/4) ✓
Smallest positive coterminal: 7π/4 rad. (Equivalent to 315°.)