Radian measure, the relationship between radians and degrees, arc length and the area of a sector and segment, and the small-angle approximations for sine, cosine and tangent.

What this dot point is asking

OCR wants you to measure angles in radians, convert between radians and degrees, recall the exact trigonometric values in radians, use $s = r\theta$ for arc length and $A = \tfrac{1}{2}r^2\theta$ for sector area (and combine them to find a segment), and apply the small-angle approximations $\sin\theta \approx \theta$, $\tan\theta \approx \theta$ and $\cos\theta \approx 1 - \tfrac{1}{2}\theta^2$ for small $\theta$ in radians.

The answer

What a radian is

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Because the full circumference is $2\pi r$, a full turn is $2\pi$ radians.

Common exact angles in radians are $30^\circ = \tfrac{\pi}{6}$, $45^\circ = \tfrac{\pi}{4}$, $60^\circ = \tfrac{\pi}{3}$, $90^\circ = \tfrac{\pi}{2}$ and $180^\circ = \pi$. From the previous topic, $\sin\tfrac{\pi}{6} = \tfrac{1}{2}$, $\cos\tfrac{\pi}{4} = \tfrac{1}{\sqrt{2}}$, $\tan\tfrac{\pi}{3} = \sqrt{3}$, and so on.

Arc length and sector area

When the angle is in radians the formulae are simple and clean (this is the main reason radians exist).

Segments

A segment is the region between a chord and its arc. Find it by subtracting the triangle (formed by the two radii and the chord) from the sector.

Small-angle approximations

When $\theta$ is small and measured in radians, the functions are close to simple polynomials. These approximations let you simplify limits and model situations where angles are tiny.

Examples in context

Mixing the formulae

A common multi-step question gives a sector and asks for the perimeter of the shaded region, which is the arc plus a chord or plus two radii. Read carefully which boundary pieces are wanted.

Why small-angle results need radians

The approximation $\sin\theta \approx \theta$ only holds when $\theta$ is in radians; in degrees it is nonsense. The same is true of the derivatives $\tfrac{d}{dx}(\sin x) = \cos x$ and $\tfrac{d}{dx}(\cos x) = -\sin x$, which assume radians. Whenever calculus or a limit meets trigonometry, work in radians.

Try this

Q1. Convert $\tfrac{5\pi}{6}$ radians to degrees. [1 mark]

• Cue. $\tfrac{5\pi}{6} \times \tfrac{180}{\pi} = 150^\circ$.

Q2. A sector has radius $10$ cm and angle $1.2$ radians. Find its area. [2 marks]

• Cue. $A = \tfrac{1}{2}(100)(1.2) = 60$ cm$^2$.