Degree-Radian Relation
$\begin{aligned} \pi \text { radians } &=180 \text { degrees } \\\\ 1 \text { radian } \times \pi &=180 \text { degrees } \\\\ 1 \text { radian } &=\displaystyle\frac{180^{\circ}}{\pi} \\\\ &=57.2957 \text { degrees } \\\\ &\approx 57^{\circ} 18^{\prime} \end{aligned}$
$\begin{aligned} 180 \text { degree } &=\pi \text { radians } \\\\ 1 \text { degree } \times 180 &=\pi \text { radians } \\\\ 1 \text { degree } &=\displaystyle\frac{\pi}{180} \text { radians } \\\\ & \approx 0.01745 \text { radians } \end{aligned}$
Central Angle, Arc Length and Area of a Sector
(i) $\theta=\displaystyle\frac{s}{r}$
(ii) $s=r \theta$
(iii) $A=\displaystyle\frac{1}{2} r^{2} \theta$
where $\theta$ must be radian measure.
Exercise (10.1)- Problems and Solutions
- Convert each of the following to radians.
(a) $120^{\circ}$ (b) $90^{\circ}$ (c) $72^{\circ}$ (d) $225^{\circ}$ (e) $150^{\circ}$ (f) $108^{\circ}$ (g) $160^{\circ}$ (h) $390^{\circ}$ - Convert each of the following to degrees.
(a) $\displaystyle\frac{\pi}{5}$ (b) $\displaystyle\frac{3 \pi}{4}$ (c) $\displaystyle\frac{5 \pi}{6}$ (d) $\pi$ (e) $\displaystyle\frac{8 \pi}{9}$ (f) $\displaystyle\frac{12 \pi}{5}$ (g) $\displaystyle\frac{\pi}{3}$ (h) $\displaystyle\frac{7 \pi}{3}$ - A central angle $\theta$ subtends an arc of $\displaystyle\frac{11 \pi}{2}$ cm on a circle of radius $6$ cm. Find the measure of $\theta$ in radians and the area of a sector of a circle which has $\theta$ is its central angle.
- The area of a sector of a circle is $143 \mathrm{~cm}^{2}$ and the length of the arc of a sector is $11\ \mathrm{~cm}$. Find the radius of the circle.
- A sector cut from a circle of radius $3 \mathrm{~cm}$ has a perimeter of $16 \mathrm{~cm}$. Find the area of this sector.
- A piece of wire of fixed length $L\ \mathrm{~cm}$, is bent to form the boundary a sector of a circle. The circle has radius $r\ \mathrm{~cm}$ and the angle of the sector is $\theta=\left(\displaystyle\frac{32}{r}-2\right)$ radians. Find the wire of fixed length $L$ and show that the area of the sector, $A \mathrm{~cm}^{2}$ is given by $A=16 r-r^{2}$.
- A race is run at a uniform speed on a circular course. In each minute, a runner traverses an arc of a circle which subtends $2 \displaystyle\frac{6}{7}$ radians at the centre of the course. If each lap is $792$ yards, how long does the runner take to run a mile?
- The large hand of a clock is $28$ inches long; how many inches does its extremity move in $20$ minutes?
- The figure shows two sectors in which the arcs $A B$ and $C D$ are arcs of concentric circles, centre $O$. If $\angle A O B=\displaystyle\frac{2}{3}$ radians, $A C=3 \mathrm{~cm}$ and the area of a sector $A O B$ is $12 \mathrm{~cm}^{2}$, calculate the area and the perimeter of $A B D C .$
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