Maths › Trigonometry › Radians, arcs and small angles

Radians, arcs and small angles

Degrees are a historical accident; radians are what circles actually want. Measure angles by arc length and the sector formulae collapse to two short products, calculus starts working, and for small angles the trig functions themselves flatten into polynomials.

Year 12-13EDEXCEL 9MA0 5.1, 5.2

Builds on Trigonometric graphs and equations.

IN THIS TOPIC

  • Convert between degrees and radians, and know the exact-value angles in both.
  • Use s = rθ and A = ½r²θ for arcs, sectors and segments.
  • Apply the small-angle approximations, in radians only.

WHAT YOU PROBABLY THINK

The small-angle approximations work in degrees too.

The natural unit

A radian is the angle whose arc equals the radius, so a full turn of circumference 2πr holds exactly 2π radians, and π radians = 180°. The familiar angles convert to fractions of π: 30° is π/6, 45° is π/4, 60° is π/3, 90° is π/2, and the exact-value table survives the move untouched.

One radian drawn on a circle: an arc exactly as long as the radius subtends an angle of one radian at the centre, about 57.3 degrees, and wrapping pi such arcs along the top half reaches 180 degreesradius rarc, also length r1 radian ≈ 57.3°the angle whose arc is one radius; π of them make 180°
FIG. 1One radian: the angle whose arc is exactly one radius, about 57.3°. Wrap π of these arcs around the top half and you have 180°.

The reward for the new unit is immediate. Arc length and sector area, both on the must-learn list, become

s = rθNOT IN THE BOOKLET — LEARN IT
A = 12r2θNOT IN THE BOOKLET — LEARN IT

with θ in radians, and with no factors of 360 anywhere in sight.

WORKED EXAMPLE

A sector, measured completely

A sector has radius 6 cm and angle 1.2 radians. Find its arc length, its perimeter, and its area.

Arc: s = 6 × 1.2 = 7.2 cm.

Perimeter: arc plus two radii, 7.2 + 12 = 19.2 cm.

Area: ½ × 36 × 1.2 = 21.6 cm².

Forgetting the two radii in the perimeter is the standard slip; a sector's boundary is an arc and two straight edges.

YOUR TURN

A segment, by subtraction

Find the area of the minor segment cut off by a chord subtending π/3 at the centre of a circle of radius 5 cm, before opening the working.

Show the working

Sector area: ½ × 25 × π/3 = 13.09 cm².

Triangle area: ½ × 25 × sin (π/3) = 10.83 cm².

Segment = sector − triangle = 2.26 cm².

Every segment question is this subtraction, and the triangle's angle is the same θ the sector used, which is why radians keep the working compact.

Small angles

Zoom in near θ = 0, in radians, and the trig curves straighten out. The specification's three approximations,

sin θ ≈ θ, cos θ ≈ 1 − θ22, tan θ ≈ θNOT IN THE BOOKLET — LEARN IT

hold because the arc and its sine become indistinguishable when the angle is tiny. The radian condition is the whole point, and the opening lie's downfall: sin 1° is 0.0175, nowhere near 1, because a degree is not the unit the approximation was built in.

The curves of sine theta, theta itself and tan theta for small positive theta in radians: all three leave the origin together and are almost indistinguishable below about 0.3, with sine drifting under the line and tan over ittan θθsin θbelow θ ≈ 0.3 the three curves are one
FIG. 2sin θ, θ and tan θ near zero. Below about 0.3 radians the three run together; further out, sine sags below the line and tan climbs above it.

TRY IT UNSEEN

A limit by approximation

For small θ, find the approximate value of (1 − cos 2θ)/(θ sin θ).

Show the working

Replace each piece: cos 2θ ≈ 1 − (2θ)2/2 = 1 − 2θ2, and sin θ ≈ θ.

The numerator becomes 2θ2 and the denominator θ2, so the expression is approximately 2.

A numeric check at θ = 0.05 gives 1.9988, and the approximations sharpen as θ shrinks. Note the doubled angle fed the cos approximation as 2θ whole; substituting the angle exactly as it appears is where these questions are won.

THE EXAM BIT

  • Check the angle mode before anything else; radian questions say so, and degree answers to radian questions score nothing.
  • s = rθ and A = ½r²θ need θ in radians; convert first, not after.
  • Sector perimeter includes the two radii; segment area is sector minus triangle with the same angle.
  • Small-angle approximations are radian-only, and saying so in words is often a mark.
  • Substitute multiples like 2θ into the approximations whole, brackets and all.

CHECK YOURSELF

An arc of length 10 cm subtends an angle θ at the centre of a circle of radius 8 cm. Find θ and the area of the sector.

Show a hint

Rearrange s = rθ first.

Show the answer

θ = s/r = 10/8 = 1.25 radians.

Area = ½ × 64 × 1.25 = 40 cm².

No degrees appeared at any point, which is the sign the formulae were being used as designed.

A radian is one radius of arc; π of them make 180°.

Arc rθ, sector ½r²θ, and for tiny radian angles sine and tan are θ itself.

CHECK YOUR PROGRESS

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  • Convert between degrees and radians, and know the exact-value angles in both.
  • Use s = rθ and A = ½r²θ for arcs, sectors and segments.
  • Apply the small-angle approximations, in radians only.

No animated video for this topic yet; these notes stand alone.