Circular measure: radian measure, arc length, area of sector, small angle approximations

Circular Measure: Radian Measure, Arc Length, Sector Area, Trigonometric Identities, Small‑Angle Approximations & Calculus Links

1. The radian – natural unit of angular measure

A radian is the angle subtended at the centre of a circle by an arc whose length is exactly equal to the radius of the circle. Consequently, the radian relates an angle directly to the geometry of the circle.

1.1 Conversion between radians and degrees

Because a full revolution is \(2\pi\) rad = 360°, the proportion

\[ \frac{\theta_{\text{rad}}}{2\pi}=\frac{\theta_{\deg}}{360} \]

gives the conversion formulae

• Radians = π × degrees / 180
• Degrees = 180 × radians / π

2. Arc length

For a circle of radius \(r\) and a central angle \(\theta\) (in radians), the length \(s\) of the intercepted arc is

\[ s = r\theta \]

Worked examples

1. Finding an arc length

   Find the arc length subtended by a \(2\ \text{rad}\) angle in a circle of radius \(5\ \text{cm}\).

   Solution: \(s = r\theta = 5\ \text{cm}\times 2 = 10\ \text{cm}\).

2. Finding the central angle from an arc length

   Given an arc length of \(12\ \text{cm}\) on a circle of radius \(3\ \text{cm}\), determine the central angle in degrees.

   Solution: \[ \theta = \frac{s}{r}= \frac{12}{3}=4\ \text{rad} \qquad\Rightarrow\qquad \theta_{\deg}= \frac{180}{\pi}\times4\approx 229.2^{\circ}. \]

3. Area of a sector

The sector is the region bounded by two radii and the intercepted arc. Its area \(A\) is proportional to the central angle:

\[ A = \frac12\,r^{2}\theta \]

Worked examples

3. Finding a sector area

   Find the area of a sector with radius \(4\ \text{m}\) and angle \(\dfrac{\pi}{3}\) radians.

   Solution: \[ A = \frac12\times4^{2}\times\frac{\pi}{3} = \frac12\times16\times\frac{\pi}{3} = \frac{8\pi}{3}\ \text{m}^{2}. \]

4. Finding the angle from a sector area

   A sector of a circle of radius \(5\ \text{cm}\) has area \(10\ \text{cm}^{2}\). Find the central angle in radians and in degrees.

   Solution: \[ \theta = \frac{2A}{r^{2}} = \frac{2\times10}{5^{2}} = \frac{20}{25}=0.8\ \text{rad} \] \[ \theta_{\deg}= \frac{180}{\pi}\times0.8\approx 45.8^{\circ}. \]

4. Trigonometric identities in radian measure

All standard trigonometric identities remain valid when the angles are expressed in radians. For example:

• \(\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta\)
• \(\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta\)
• \(\sin2\alpha = 2\sin\alpha\cos\alpha,\qquad \cos2\alpha = \cos^{2}\alpha-\sin^{2}\alpha\)

These formulas are used throughout Pure Mathematics 1; the only requirement is that the arguments \(\alpha,\beta\) are in radians.

Example

Show that \(\sin(\pi/6)=\frac12\) using the double‑angle identity.

Solution: \[ \sin\frac{\pi}{6}= \sin\!\left(\frac{\pi}{3}\times\frac12\right) = 2\sin\frac{\pi}{12}\cos\frac{\pi}{12} \] Evaluating the known values \(\sin\frac{\pi}{6}=0.5\) confirms the identity (the step illustrates that the identity works with radian arguments).

5. Small‑angle approximations (angles in radians)

For sufficiently small angles (typically \(|\theta|<0.1\) rad) the first‑order Maclaurin expansions give:

• \(\displaystyle \sin\theta \approx \theta\)
• \(\displaystyle \tan\theta \approx \theta\)
• \(\displaystyle \cos\theta \approx 1-\frac{\theta^{2}}{2}\)

Maclaurin series (truncated after the first non‑zero term) are:

\[ \sin\theta = \theta - \frac{\theta^{3}}{6}+O(\theta^{5}),\qquad \tan\theta = \theta + \frac{\theta^{3}}{3}+O(\theta^{5}),\qquad \cos\theta = 1 - \frac{\theta^{2}}{2} + \frac{\theta^{4}}{24}+O(\theta^{6}). \]

Application example

Approximate \(\sin 0.05\) rad.

Solution: \(\sin 0.05 \approx 0.05\). Calculator value: \(0.0499792\); absolute error \(2.08\times10^{-5}\) (< 0.05 %).

6. Why radians matter in calculus

The differentiation formulas for the trigonometric functions are valid only when the angle is measured in radians. In particular:

• \(\displaystyle \frac{d}{dx}\sin x = \cos x\)
• \(\displaystyle \frac{d}{dx}\cos x = -\sin x\)

These results follow from the limit \(\displaystyle \lim_{h\to0}\frac{\sin h}{h}=1\), which holds exclusively for radian measure. (A full proof is given later in the calculus chapter, but the key point for the syllabus is the dependence on radians.)

7. Links to other Pure 1 topics

• Period of sine and cosine: \(\sin x\) and \(\cos x\) repeat every \(2\pi\) rad (i.e., period = \(2\pi\)).
• Solving simple trigonometric equations (example):

  Find all solutions of \(\sin x = \tfrac12\) in the interval \([0,2\pi)\).

  Solution: \(\displaystyle x = \frac{\pi}{6}\) or \(x = \frac{5\pi}{6}\) (both in radians).

• Sketching graphs: When drawing the graph of \(\sin x\) or \(\cos x\), mark key points at multiples of \(\frac{\pi}{2}\) (e.g., \(0,\ \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ 2\pi\)).

8. Summary of key formulae