The approximate value of \(\cos \left(30^{\circ}, 30^{\prime}\right)\) is, given that \(1^{\circ}=0.0175^{\circ}\) and \(\cos 30^{\circ}=0.8660\)

\(\begin{aligned} & \cos \left(30^{\circ}, 30^{\prime}\right) \\ & \text { Let } \mathrm{f}(x)=\cos x \\ \therefore \quad & \mathrm{f}^{\prime}(x)=-\sin x \\ \therefore \quad & \mathrm{f}^{\prime}(x)=-\sin x \\ & \text { Now, } 30^{\circ} 30^{\prime}=30^{\circ}+30^{\prime} \\ & =30^{\circ}+\left(\frac{1}{2}\right)^{\circ} \\ & \\ = & 30^{\circ}+\frac{0.0175}{2} \\ \therefore \quad & 30^{\circ} 30^{\prime}=30^{\circ}+0.00875\end{aligned}\)
\(\begin{aligned} & \text { Let } \mathrm{a}=30^{\circ} \text { and } \mathrm{h}=0.00875 \\ & \therefore \quad \cos \left(30^{\circ}, 30^{\prime}\right)=\mathrm{f}(\mathrm{a}+\mathrm{h}) \\ & \approx \mathrm{f}(\mathrm{a})+\mathrm{h} \mathrm{f}^{\prime}(\mathrm{a}) \\ & \approx \mathrm{f}\left(30^{\circ}\right)+0.00875 \mathrm{f}^{\prime}\left(30^{\circ}\right) \\ & \approx \cos \left(30^{\circ}\right)-0.00875 \times \sin \left(30^{\circ}\right) \\ & \approx 0.8660-0.00875 \times 0.5 \\ & \approx 0.8616\end{aligned}\)