Radians, degrees & arc length
One radian is the angle subtending an arc equal to the radius, so a full turn is 2π radians and 180° = π. The reason radians dominate in calculus is that the clean formulas s = rθ and A = ½r²θ, and the derivative d/dx sin x = cos x, all require radian measure — in degrees they pick up ugly constants.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Convert with the ratio π/180Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π.
- Use s = rθ for arc lengthθ MUST be in radians. This is the single most common source of error.
- Use A = ½r²θ for sector areaAgain radians only. Check against the full circle: θ = 2π gives πr² ✓.
- Keep π symbolicExact answers like 5π/6 are usually required, not 2.618.
Worked example
A sector has radius 6 cm and angle 60°. Find its arc length and area.
- Convert: 60° × π/180 = π/3 radians.
- Arc length: s = rθ = 6 × π/3 = 2π ≈ 6.28 cm.
- Area: A = ½r²θ = ½ × 36 × π/3.
- = 18 × π/3 = 6π ≈ 18.85 cm².
Answer. Arc length 2π cm and area 6π cm² — one sixth of the circle’s circumference and area, as 60° is one sixth of a turn ✓.
Where marks get dropped
These are the specific errors that cost credit on radians, degrees & arc length questions — QED's rubric penalises each of them separately.
- Using degrees in s = rθ or A = ½r²θ. Both formulas are radian-only and give answers wrong by a factor of about 57.
- Converting the wrong way. Multiply degrees by π/180 — the radian answer should be numerically smaller.
- Confusing the sector area formula with the triangle area ½ab sin C. The sector is bounded by an arc, not a chord.
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Radians, degrees & arc length — frequently asked questions
Why use radians at all?
Because they make the calculus clean: d/dx sin x = cos x only holds in radians, and lim(sin x/x) = 1 likewise. In degrees both acquire a factor of π/180.
How many degrees is one radian?
180/π ≈ 57.3°. A useful check: an answer in radians should be roughly 1/57 of the degree value.
What is the area of a segment?
Sector minus triangle: ½r²θ − ½r² sin θ = ½r²(θ − sin θ), with θ in radians.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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