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°.)