方程 $x\sin x + \cos x = 1.015$ 有一个接近于零的正根 $\alpha$。利用 $\sin x$ 和 $\cos x$ 的小角近似，求出 $\alpha$ 的近似值。答案保留 3 位有效数字。

1. Substitute the small-angle approximations for $\sin x$ and $\cos x$ into the given equation.

Using $\sin x \approx x$ and $\cos x \approx 1 - \frac{1}{2}x^2$ for small $x$ (in radians):
$x(x) + \left(1 - \frac{1}{2}x^2\right) \approx 1.015$

2. Simplify the resulting algebraic expression to isolate $x^2$.

$x^2 + 1 - 0.5x^2 = 1.015$
$0.5x^2 + 1 = 1.015$
$0.5x^2 = 0.015$

3. Solve for the positive root and round to 3 significant figures.

$x^2 = \frac{0.015}{0.5} = 0.03$
Since $\alpha$ is a positive root, we take the positive square root:
$x = \sqrt{0.03} \approx 0.173205$
$\alpha = 0.173 \text{ (to 3 s.f.)}$