1. Problem: Find, in terms of $\theta$, an approximate value for $$\frac{\sin 4\theta + \tan 2\theta}{3 + \cos 2\theta}$$ for small values of $\theta$ (i.e., $\theta \to 0$) neglecting terms of order $\theta^2$ and higher. 2. Recall the small-angle approximations valid for $\theta \to 0$ (in radians): - $\sin x \approx x$ - $\tan x \approx x$ - $\cos x \approx 1$ 3. Apply these to each term: - $\sin 4\theta \approx 4\theta$ - $\tan 2\theta \approx 2\theta$ - $\cos 2\theta \approx 1$ 4. Substitute into the expression: $$\frac{\sin 4\theta + \tan 2\theta}{3 + \cos 2\theta} \approx \frac{4\theta + 2\theta}{3 + 1} = \frac{6\theta}{4} = \frac{3\theta}{2}$$ 5. So, the simplified approximation is: $$\boxed{\frac{3\theta}{2}}$$ --- 6. Next, find an approximate value for $$\frac{\cos^2 1^\circ}{\sin 1^\circ}$$ given: - $1^\circ \approx 0.018$ radians - $(0.018)^2 \approx 0.000324$ 7. Using small-angle approximations: - $\sin 1^\circ \approx 0.018$ - $\cos 1^\circ \approx 1 - \frac{(0.018)^2}{2} = 1 - \frac{0.000324}{2} = 1 - 0.000162 = 0.999838$ 8. Then: $$\cos^2 1^\circ \approx (0.999838)^2 \approx 0.9997$$ (neglecting higher powers) 9. Now compute: $$\frac{\cos^2 1^\circ}{\sin 1^\circ} \approx \frac{0.9997}{0.018} \approx 55.54$$ 10. To 2 decimal places, the answer is: $$\boxed{55.54}$$