题目

Problem

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(a) Solve, for $0 < \theta \leqslant 360°$ , the equation

$2\tan\theta + 3\sin\theta = 0$

giving your answers, as appropriate, to one decimal place.

(b) Hence, or otherwise, find the smallest positive solution of

$2\tan(2x + 40°) + 3\sin(2x + 40°) = 0$

giving your answer to one decimal place.

(7)

题目中文翻译

本题必须展示所有解题步骤，不能完全依赖计算器技术。

(a) 解方程 $2\tan\theta + 3\sin\theta = 0$ ，其中 $0 < \theta \leqslant 360°$ ，答案保留一位小数。

(b) 由此或 otherwise，求 $2\tan(2x + 40°) + 3\sin(2x + 40°) = 0$ 的最小正解，答案保留一位小数。

（7分）

解答

将 $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ 代入，提取公因子 $\sin\theta$ ，分别解 $\sin\theta = 0$ 和 $\cos\theta = -\dfrac{2}{3}$ 。

(a) Solve $2\tan\theta + 3\sin\theta = 0$ for $0 < \theta \leqslant 360°$ :

2 \cdot \frac{\sin\theta}{\cos\theta} + 3\sin\theta =&\, 0 \\[2mm] \sin\theta \left(\frac{2}{\cos\theta} + 3\right) =&\, 0 \\[2mm] \sin\theta \left(\frac{2 + 3\cos\theta}{\cos\theta}\right) =&\, 0 \end{align*}$$ So $\sin\theta = 0$ or $2 + 3\cos\theta = 0$. **Case 1:** $\sin\theta = 0$ $$\theta = 180°, \quad 360° \qquad (\theta = 0° \text{ excluded since } 0 < \theta)$$ **Case 2:** $\cos\theta = -\dfrac{2}{3}$ $$\theta = 131.8°, \quad 228.2°$$ $$\boxed{\theta = 131.8°, \quad 180°, \quad 228.2°, \quad 360°}$$ **(b)** Let $\phi = 2x + 40°$. The equation becomes $2\tan\phi + 3\sin\phi = 0$. From part (a), the smallest positive solution is $\phi = 131.8°$: $$\begin{align*} 2x + 40° =&\, 131.8° \\ 2x =&\, 91.8° \\ x =&\, 45.9° \end{align*}$$ $$\boxed{x = 45.9°}$$