Cambridge IGCSE Additional Mathematics · Syllabus 0606 · Circular Measure

Radian

What is Radian?

The unit of angle defined by the ratio of the arc a central angle cuts off to the radius of the circle, so that an angle of one radian is the angle for which the arc length is exactly equal to the radius, approximately 57.3 degrees. Because it is one length divided by another, an angle in radians is a pure number carrying no physical unit, which is why the arc-length and sector-area formulas contain no conversion constant and are valid only when the angle is measured this way.

This definition is part of the Circular Measure chapter in Cambridge IGCSE Additional Mathematics.

Common mistakes with Radian

Taking a sine in degree mode on a radian angle. Why it fails \(\sin 1.2\) means the sine of \(1.2\) radians, which is \(0.932\). A calculator in degree mode returns \(\sin 1.2^\circ=0.0209\) — the sine is out by a factor of about \(45\). The segment answer that follows is not out by \(45\); it is out by rather less, because the sine sits inside a subtraction. For \(r=8\), \(\theta=1.2\) it returns \(37.7\ \mathrm{cm^2}\) instead of \(8.57\ \mathrm{cm^2}\), about four times too large; at other angles the factor is different again. What is constant is not the size of the error but its shape: the triangle almost vanishes, so the segment comes out as nearly the whole sector. Fix Set radian mode at the start of the topic. Two sanity checks: for \(0<\theta<\pi\), \(\sin\theta\) should be a healthy fraction, not something starting \(0.0\); and a segment should be a modest slice of its sector, never \(98\%\) of it.
Using \(s=r\theta\) with \(\theta\) in degrees. Why it fails \(s=r\theta\) is the definition \(\theta=s/r\) rearranged, and that definition measures the angle in radii of arc. A degree is a different and much smaller unit, so a degree value is about \(57.3\) times the radian value and the arc comes out about \(57.3\) times too long. Fix Convert on a separate line first. Then check the arc against \(2\pi r\); an arc longer than the circumference is impossible.
Using \(\dfrac{\pi r^2\theta}{360}\) while \(\theta\) has already been converted to radians. Why it fails This is the degree formula fed a radian value. The \(360\) in the denominator is only correct if the numerator angle is measured in degrees; with radians the correct denominator is \(2\pi\). The answer comes out about \(57.3\) times too small. Fix Choose one system before writing anything. In radians: \(\tfrac12r^2\theta\). In degrees: \(\dfrac{\theta}{360}\times\pi r^2\). Never a hybrid.
Forgetting that \(180^\circ=\pi\) radians, and reaching instead for “\(360=\pi\)” or “\(90=\pi\)”. Why it fails \(\pi\) radians is a half turn, because the full turn is \(2\pi\) — which is itself a consequence of the circumference containing \(2\pi\) radii. Fix Anchor on the right angle instead, which is harder to misremember: \(90^\circ=\pi/2\). Doubling it gives \(180^\circ=\pi\).
Converting degrees to radians by multiplying by \(\dfrac{180}{\pi}\). Why it fails That multiplier is about \(57.3\), so it makes the number bigger — but radians are the larger unit, so a given angle has fewer of them and the number must get smaller. Fix Test the multiplier on \(90^\circ\). It must produce \(\pi/2\approx1.57\), not \(5157\).
Converting radians to degrees by multiplying by \(\dfrac{\pi}{180}\). Why it fails The mirror image of the previous entry: that multiplier is about \(0.0175\) and shrinks the number, whereas degrees are the smaller unit and there must be more of them. Fix Same test. \(\pi/2\) must come out as \(90\).
Taking a sine in degree mode on a radian angle. Why it fails \(\sin1.2\) asks for the sine of \(1.2\) radians, roughly \(69^\circ\), which is \(0.932\). In degree mode the calculator returns \(\sin1.2^\circ=0.0209\), the sine of a nearly flat angle. Every segment computed from it is wrong by a wide margin. Fix Set RAD at the start of the topic. Recognise the signature: a sine near \(0.02\) for an angle that ought to be substantial.

Examiner tips on Radian

The one-line reminder to carry forward. Write it on the inside cover of your notes: radians, then radius, then trace the boundary. Those three steps in that order head off the angle-unit and boundary errors that make up most of the mistake clinic, and none of them requires remembering a formula.

What is Radian?

Common mistakes with Radian

Taking a sine in degree mode on a radian angle. Why it fails \(\sin 1.2\) means the sine of \(1.2\) radians, which is \(0.932\). A calculator in degree mode returns \(\sin 1.2^\circ=0.0209\) — the sine is out by a factor of about \(45\). The segment answer that follows is not out by \(45\); it is out by rather less, because the sine sits inside a subtraction. For \(r=8\), \(\theta=1.2\) it returns \(37.7\ \mathrm{cm^2}\) instead of \(8.57\ \mathrm{cm^2}\), about four times too large; at other angles the factor is different again. What is constant is not the size of the error but its shape: the triangle almost vanishes, so the segment comes out as nearly the whole sector. Fix Set radian mode at the start of the topic. Two sanity checks: for \(0<\theta<\pi\), \(\sin\theta\) should be a healthy fraction, not something starting \(0.0\); and a segment should be a modest slice of its sector, never \(98\%\) of it.

Using \(s=r\theta\) with \(\theta\) in degrees. Why it fails \(s=r\theta\) is the definition \(\theta=s/r\) rearranged, and that definition measures the angle in radii of arc. A degree is a different and much smaller unit, so a degree value is about \(57.3\) times the radian value and the arc comes out about \(57.3\) times too long. Fix Convert on a separate line first. Then check the arc against \(2\pi r\); an arc longer than the circumference is impossible.

Using \(\dfrac{\pi r^2\theta}{360}\) while \(\theta\) has already been converted to radians. Why it fails This is the degree formula fed a radian value. The \(360\) in the denominator is only correct if the numerator angle is measured in degrees; with radians the correct denominator is \(2\pi\). The answer comes out about \(57.3\) times too small. Fix Choose one system before writing anything. In radians: \(\tfrac12r^2\theta\). In degrees: \(\dfrac{\theta}{360}\times\pi r^2\). Never a hybrid.

Forgetting that \(180^\circ=\pi\) radians, and reaching instead for “\(360=\pi\)” or “\(90=\pi\)”. Why it fails \(\pi\) radians is a half turn, because the full turn is \(2\pi\) — which is itself a consequence of the circumference containing \(2\pi\) radii. Fix Anchor on the right angle instead, which is harder to misremember: \(90^\circ=\pi/2\). Doubling it gives \(180^\circ=\pi\).

Converting degrees to radians by multiplying by \(\dfrac{180}{\pi}\). Why it fails That multiplier is about \(57.3\), so it makes the number bigger — but radians are the larger unit, so a given angle has fewer of them and the number must get smaller. Fix Test the multiplier on \(90^\circ\). It must produce \(\pi/2\approx1.57\), not \(5157\).

Converting radians to degrees by multiplying by \(\dfrac{\pi}{180}\). Why it fails The mirror image of the previous entry: that multiplier is about \(0.0175\) and shrinks the number, whereas degrees are the smaller unit and there must be more of them. Fix Same test. \(\pi/2\) must come out as \(90\).

Taking a sine in degree mode on a radian angle. Why it fails \(\sin1.2\) asks for the sine of \(1.2\) radians, roughly \(69^\circ\), which is \(0.932\). In degree mode the calculator returns \(\sin1.2^\circ=0.0209\), the sine of a nearly flat angle. Every segment computed from it is wrong by a wide margin. Fix Set RAD at the start of the topic. Recognise the signature: a sine near \(0.02\) for an angle that ought to be substantial.

What is Radian?

Concepts related to Radian

Common mistakes with Radian

Examiner tips on Radian

What is Radian?

Concepts related to Radian

Common mistakes with Radian

Examiner tips on Radian