Circular Measure Revision Notes — 9709 Paper 1

Radians

Syllabus requirement · §1.4

“

understand the definition of a radian, and use the relationship between radians and degrees.

”

Why degrees have to go

Degrees are arbitrary. Somebody, a long time ago, decided that a full turn should be cut into $360$ equal pieces — probably because $360$ has a lot of factors and because a year is roughly $360$ days. There is nothing about a circle that makes $360$ special. You could just as well have chosen $100$ , or $17$ .

That arbitrariness has a price. If you measure the angle $\theta$ in degrees, the arc it cuts off is

s = \frac{\theta}{360} \times 2\pi r = \frac{\pi}{180}\, r\theta

and you are stuck carrying that ugly $\frac{\pi}{180}$ around forever. Radians are the unit that makes the conversion factor equal to $1$ .

The definition

Take a circle of radius $r$ . Bend a piece of string of length exactly $r$ around the circumference. The angle at the centre that the string subtends is defined to be one radian.

That is the whole definition. It says nothing about $360$ , and it does not depend on how big the circle is: a circle twice as wide needs a string twice as long, and the angle comes out the same.

Bend a length equal to the radius around the circumference and the angle you have swept out is one radian. Since the whole circumference is 2πr — that is, 2π radius-lengths — a full turn is 2π radians.

Deriving the conversion

Now count how many radius-lengths fit round the whole circle. The circumference is $2\pi r$ , and each radian uses up one radius-length of arc, so

\text{a full turn} = \frac{2\pi r}{r} = 2\pi \text{ radians}.

A full turn is also $360^\circ$ . So $2\pi \text{ rad} = 360^\circ$ , and halving both sides gives the only conversion you will ever need.

\pi \text{ rad} = 180^\circ

The only conversion you need

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

Doing it on clean numbers

Degrees to radians. Multiply by $\dfrac{\pi}{180}$ . Convert $135^\circ$ :

135 \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{27\pi}{36} = \frac{3\pi}{4}

Cancel before you reach for a calculator — the answer is almost always meant to be a neat fraction of $\pi$ , and $\frac{3\pi}{4}$ is a better answer than $2.356$ .

Radians to degrees. Multiply by $\dfrac{180}{\pi}$ . Convert $\dfrac{5\pi}{6}$ :

\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150^\circ

The $\pi$ cancels, which is the whole point of writing radians as multiples of $\pi$ in the first place.

A decimal angle. Convert $1.4$ rad to degrees:

1.4 \times \frac{180}{\pi} = 1.4 \times 57.2957\ldots = 80.2^\circ \ (\text{1 d.p.})

so one radian is about $57.3^\circ$ . Keep that number in your head as a sanity check: an angle of $2$ radians is a bit under $115^\circ$ , so it is obtuse; an angle of $4$ radians is reflex.

Degrees	Radians	Where it turns up in this topic
30°	$\frac{\pi}{6}$	exact-value questions; half of an equilateral triangle
45°	$\frac{\pi}{4}$	squares and isosceles right-angled triangles
60°	$\frac{\pi}{3}$	equal radii forcing an equilateral triangle
90°	$\frac{\pi}{2}$	tangents, perpendiculars, quarter circles
120°	$\frac{2\pi}{3}$	three equal sectors; chord $= r\sqrt3$
135°	$\frac{3\pi}{4}$	the supplement of 45°
150°	$\frac{5\pi}{6}$	the supplement of 30°
180°	$\pi$	semicircles; angles on a straight line
360°	$2\pi$	full turns; reflex and major angles

Worth knowing on sight — questions hand you an angle like 2π⁄3 with no comment and expect you to know it is 120°.

Exact values, in radians

Many answers in this topic are meant to come out exactly, in terms of $\pi$ and $\sqrt3$ , so you need the O Level exact values written with radian angles. For the obtuse angles, use the supplement rules $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$ (the same as $\sin(180^\circ - x) = \sin x$ from O Level):

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$
$\frac{\pi}{6}$	$\frac12$	$\frac{\sqrt3}{2}$	$\frac{1}{\sqrt3}$
$\frac{\pi}{4}$	$\frac{1}{\sqrt2}$	$\frac{1}{\sqrt2}$	$1$
$\frac{\pi}{3}$	$\frac{\sqrt3}{2}$	$\frac12$	$\sqrt3$
$\frac{\pi}{2}$	$1$	$0$	undefined
$\frac{2\pi}{3}$	$\frac{\sqrt3}{2}$	$-\frac12$	$-\sqrt3$

The last row is the one students miss. sin(2π⁄3) = sin(π⁄3) because 2π⁄3 is the supplement of π⁄3 — but the cosine changes sign.

Set your calculator to radians

If your calculator is in degree mode, $\sin 1.2$ or $\tan^{-1}\tfrac13$ silently gives a wrong number that looks plausible — there is no error message. Check the mode indicator (R or RAD) before you start.

A quick test: $\sin(1)$ should be $0.841$ . If your calculator says $0.0175$ , you are in degrees.

In the exam

Radians on their own: only 7 parts · 20 marks, 2021–2025

Recent papers do not ask you to "convert $150^\circ$ to radians" on its own. Radians turn up as an opener such as "Show that angle $ABC = 1.25$ radians, correct to 3 significant figures" — a $1$ – $2$ mark part that gives you the angle for the rest of the question, provided you can get it out of the diagram. The section "Getting the angle and the radius out of the diagram" shows how.

Your turn

Nothing here needs a diagram. Get the conversions automatic so that in the exam they cost you no thought at all.

1
Convert to radians, giving each answer as an exact multiple of $\pi$ :
(a) $210^\circ$ (b) $22.5^\circ$

Convert to degrees:
(c) $\dfrac{3\pi}{8}$ (d) $2.5$ radians, correct to 1 decimal place.
Show solution
1. 1
  (a) Multiply by $\dfrac{\pi}{180}$ : $210 \times \frac{\pi}{180} = \frac{210\pi}{180}$
  Write the fraction before cancelling; it is much easier to see the common factor with the numbers side by side.
2. 2
  Both $210$ and $180$ divide by $30$ : $\dfrac{210}{180} = \dfrac{7}{6}$ , so the angle is $\dfrac{7\pi}{6}$ .
3. 3
  (b) $22.5 \times \dfrac{\pi}{180} = \dfrac{22.5\pi}{180}$ . Multiply top and bottom by $2$ to clear the decimal: $\dfrac{45\pi}{360}$ .
  Decimals inside a fraction hide the cancelling. Clear them first.
4. 4
  $\dfrac{45}{360} = \dfrac{1}{8}$ , so the angle is $\dfrac{\pi}{8}$ .
5. 5
  (c) Multiply by $\dfrac{180}{\pi}$ : $\frac{3\pi}{8} \times \frac{180}{\pi} = \frac{3 \times 180}{8} = \frac{540}{8} = 67.5^\circ$
  The π cancels immediately — that is why you leave radian answers in terms of π wherever you can.
6. 6
  (d) Multiply by $\dfrac{180}{\pi}$ : $2.5 \times \frac{180}{\pi} = \frac{450}{\pi} = 143.239\ldots = 143.2^\circ$
  Sanity check: 2.5 radians is between π/2 ≈ 1.57 and π ≈ 3.14, so the angle must be obtuse. 143.2° is.
Answer
(a) $\dfrac{7\pi}{6}$ (b) $\dfrac{\pi}{8}$ (c) $67.5^\circ$ (d) $143.2^\circ$
2
Write down the exact value of each, without a calculator:
(a) $\sin\dfrac{2\pi}{3}$ (b) $\cos\dfrac{3\pi}{4}$ (c) $\tan\dfrac{\pi}{6}$ (d) $\sin\dfrac{\pi}{2}$
Stuck? Show hint
Convert each angle to degrees in your head first. $\dfrac{2\pi}{3}$ is $120^\circ$ , an obtuse angle: use $\sin(180^\circ - x) = \sin x$ or $\cos(180^\circ - x) = -\cos x$ .
Show solution
1. 1
  (a) $\dfrac{2\pi}{3} = 120^\circ$ , which is obtuse. Its supplement is $180^\circ - 120^\circ = 60^\circ$ , and an angle and its supplement have the same sine: $\sin\frac{2\pi}{3} = \sin 60^\circ = \frac{\sqrt3}{2}$
2. 2
  (b) $\dfrac{3\pi}{4} = 135^\circ$ , also obtuse, with supplement $45^\circ$ . The cosine of an obtuse angle is negative: $\cos\frac{3\pi}{4} = -\cos 45^\circ = -\frac{1}{\sqrt2}$
  The minus sign is the whole question here. The sine of an obtuse angle is positive; its cosine is negative.
3. 3
  (c) $\dfrac{\pi}{6} = 30^\circ$ , and $\tan 30^\circ = \dfrac{1}{\sqrt3}$ (equivalently $\dfrac{\sqrt3}{3}$ ).
4. 4
  (d) $\dfrac{\pi}{2} = 90^\circ$ and $\sin 90^\circ = 1$ .
Answer
(a) $\dfrac{\sqrt3}{2}$ (b) $-\dfrac{1}{\sqrt2}$ (c) $\dfrac{1}{\sqrt3}$ (d) $1$

Practise radian questionsReal past-paper questions · Radians and conversion between radians and degrees

Checking your access…

Can you do all of these?

Convert $135^\circ$ to radians and $\tfrac{5\pi}{6}$ to degrees without a calculator
Rebuild $s = r\theta$ and $A = \tfrac12 r^2\theta$ from " $\theta$ over $2\pi$ of the whole circle"
Find the arc length, sector area and sector perimeter for $r = 7$ , $\theta = 1.2$
Find $\theta$ given that a sector of radius $6$ has area $30$
Turn "perimeter $65$ , area $225$ " into a quadratic in $r$ , solve it, and reject the wrong root
Say what $\sin\tfrac{2\pi}{3}$ and $\cos\tfrac{3\pi}{4}$ are, exactly, without a calculator
Find a central angle from a chord using $\sin\tfrac{\theta}{2} = \tfrac{c}{2r}$ — and remember to double
Get an angle by subtracting from a right angle, a straight line or a full turn
Use the tangent–radius right angle to find a length or an angle
Identify the radius correctly when the obvious number in the diagram is not it
Derive $\tfrac12 r^2(\theta - \sin\theta)$ from "sector $-$ triangle" rather than memorising it
Write down the perimeter of a sector and of a segment, and say why they differ
Find a major segment as $\pi r^2 - {}$ the minor one
Split a shaded region into named pieces in words before computing anything
Trace a boundary to get a perimeter, ignoring internal construction lines
Find the area between two concentric arcs, and its perimeter (the straight edges are $R - r$ )
Give an area exactly in terms of $\pi$ , $\sqrt3$ and $r^2$ , with no decimals anywhere in the working
Check a symbolic answer dimensionally: lengths carry $r$ , areas carry $r^2$

Now do the questions

68 real Paper 1 parts from 2021–2025, sorted by difficulty, with mark schemes

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