3.7 Radian measureIB Maths: Applications and Interpretation HL: Revision notes

One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. A full turn has arc length $2\pi r$, so $2\pi\text{ radians}=360^\circ,\qquad\pi\text{ radians}=180^\circ.$ To convert: degrees $\to$ radians, multiply by $\frac{\pi}{180}$; radians $\to$ degrees, multiply by $\frac{180}{\pi}$. Exact values to know: $30^\circ=\frac\pi6$, $45^\circ=\frac\pi4$, $60^\circ=\frac\pi3$, $90^\circ=\frac\pi2$. Examples: $150^\circ=\frac{5\pi}{6}$, and $2.4\text{ rad}=2.4\times\frac{180}{\pi}=138^\circ$. A radian measure can be written as a multiple of $\pi$ or as a decimal.

For a sector with radius $r$ and angle $\theta$ in radians: $l=r\theta.$ This comes from $\frac{\theta}{2\pi}$ of the circumference $2\pi r$. Example: $r=6$ cm, $\theta=\frac{2\pi}{3}$: $l=6\times\frac{2\pi}{3}=4\pi$ cm. The perimeter of a sector is $2r+r\theta$: here $12+4\pi=24.6$ cm.

With $\theta$ in radians: $A=\frac12r^2\theta.$ This is $\frac{\theta}{2\pi}$ of the circle's area $\pi r^2$. Using $l=r\theta$ you can also write $A=\frac12rl$. Example: $r=6$, $\theta=\frac{2\pi}{3}$: $A=\frac12(36)\left(\frac{2\pi}{3}\right)=12\pi$ cm$^2$. Example: arc length 15 cm and area 75 cm$^2$: $\frac12r(15)=75$ gives $r=10$ and $\theta=\frac{15}{10}=1.5$.

Many questions combine these results.

• Part of a sector between two radii $r<R$ with the same angle: $\frac12\theta(R^2-r^2)$.
• Region between a chord and an arc: sector area minus triangle area, $\frac12r^2\theta-\frac12r^2\sin\theta$.
• Speed along an arc: distance $r\theta$ divided by time. Example: $r=8$ cm, $\theta=1.2$: sector $=38.4$ cm$^2$, triangle $=\frac12(64)\sin1.2=29.8$ cm$^2$, so the region between chord and arc is $38.4-29.8=8.57$ cm$^2$.

On HL papers, assume radians unless the question shows a degree sign. Before every trigonometric calculation, check that the GDC is in radian mode: for instance $\sin1.5=0.997$ in radians but $0.0262$ in degrees. Radians are the natural unit for the trigonometric functions of AHL 2.9: $\sin x$ and $\cos x$ have period $2\pi$. Keep exact multiples of $\pi$ when the question asks for exact values, and otherwise give 3 significant figures.