Problem
(25) The diagram to the right shows a sector $A B C$ of a circle, where the angle BAC is 0.9 radians. Given that the arca of the sector is $16.2 \mathrm{~cm}^{2}$, find the arc length $s$. (Q6) A circle of radius $r$ contains a sector of area $80 \pi \mathrm{cm}^{2}$. Given that the arc length of the sector is $16 \pi \mathrm{cm}$, find the angle of the sector $(\theta)$ and the value of $r$, giving your answers to 3 s.f. (77) The diagram to the left shows a semicircle of radius $2 \mathrm{~cm}$, with a smaller sector of radius $1 \mathrm{~cm}$ removed. Given that the area of the sector $A$ and the area of $B$ are equal, find the exact value of $\theta$. Q8 Use the small angle approximations to estimate the following values, then calculate the actual values: (a) $\sin 0.23$ b) $\sin 0.12$ (c) $\cos 0.01$ d) $\cos 0.24$ (e) $\tan 0.18$ Q9. For the values of $\theta$ below, use the small angle approximations to estimate the value of $f(\theta)=\sin \theta+\cos \theta$, then use a calculator to find the actual answer: (a) $\theta=0.3$ b) $\theta=0.5$ (c) $\theta=0.25$ d) $\theta=0.01$ (e) $\theta=0.03$ (Q10) Find an approximation for the following if $\theta$ is small: a) $\sin \theta \cos \theta$ b) $\theta \tan 5 \theta \sin \theta$ c) $\frac{\sin 4 \theta \cos 3 \theta}{2 \theta}$ d) $3 \tan \theta+\cos 2 \theta$ e) $\sin \frac{1}{2} \theta-\cos \theta$ f) $\frac{\cos \theta-\cos 2 \theta}{1-(\cos 3 \theta+3 \sin \theta \tan \theta)}$ (Qi1) A pendulum of length $6 \mathrm{~cm}$ follows the arc of a circle. Its straight-line displacement as a vector is given by: $\mathbf{d}=6 \sin \theta \mathbf{i}+6(1-\cos \theta) \mathbf{j}$. a) Show that the magnitude of the displacement is $6 \sqrt{2(1-\cos \theta)}$. b) Show that, when $\theta$ is small, the magnitude of the displacement can be approximated by the arc length $s$. (12) Solve the following guabrans for the unorals gwan $(x$ \& 3 sf $)$ : (a) $4 \cos x-3=0 \quad 0 \leqslant x \leqslant 2 \pi$ (b) $\sin 3 x=-\frac{1}{\sqrt{2}}$ for $-\pi \leqslant x \leqslant \pi$ (c) $5 \cos ^{2} x-9 \sin x=3 \quad 0 \leqslant x \leqslant 2 \pi$
Solution
Step 1 ：Let the radius of the circle be $r$ cm and the arc length be $s$ cm.
Step 2 ：The area of the sector is given by $\frac{1}{2}r^2\theta = 16.2$.
Step 3 ：Since the angle BAC is 0.9 radians, we have $\frac{1}{2}r^2(0.9) = 16.2$.
Step 4 ：Solving for $r^2$, we get $r^2 = \frac{16.2}{0.45} = 36$.
Step 5 ：Now, we know that $r = 6$ cm.
Step 6 ：The arc length $s$ can be found using the formula $s = r\theta$.
Step 7 ：Substituting the values, we get $s = 6(0.9)$.
Step 8 ：Therefore, the arc length is $s = \boxed{5.4}$ cm.
