Solutions

i)

Let

$$\delta_1 = \sin^2\left(\frac{9\epsilon^2}{4}\right)+\frac{\epsilon}{2} \qquad\text{and}\qquad \delta_2 = \frac{\epsilon^2}{4}.$$

Then, $\delta = \min(\delta_1,\delta_2)$.

ii)

Let

$$\delta_1 = \min\left(\sin^2\left(\frac{1}{4}\right)+1,\; \sin^2\left(\frac{9\epsilon^2}{100}\right)+\frac{\epsilon}{10}\right) \qquad\text{and}\qquad \delta_2 = \frac{\epsilon^2}{36}.$$

Then, $\delta = \min(\delta_1,\delta_2)$.

iii)

$$\delta = \min(4, 64\epsilon^2).$$

iv)

Let

$$\delta_1 = \min\left(\sin^2\left(\frac{1}{9}\right)+1,\; \sin^2\left(\frac{9\epsilon^2}{25}\right)+\frac{4\epsilon}{10}\right) \qquad\text{and}\qquad \delta_2 = \min(4, 16\epsilon^2).$$

Then, $\delta = \min(\delta_1,\delta_2)$.