Mastering Trigonometric Equations | 精通三角函数方程

📚 Mastering Trigonometric Equations | 精通三角函数方程

Trigonometric equations appear throughout Edexcel A-Level Pure Mathematics, especially in AS Pure 1 and A2 Pure 2. Many students lose marks not because they cannot solve the basic equations, but because they forget the periodic nature of sine, cosine and tangent, or they fail to select the correct solutions within a given interval.

三角函数方程贯穿 Edexcel A-Level 纯数学的各个部分，尤其是在 AS Pure 1 和 A2 Pure 2 中。许多学生失分并不是因为不会解基本方程，而是因为他们忘记了正弦、余弦和正切的周期性，或者未能在指定区间内选出正确的解。

1. Understanding the Unit Circle | 理解单位圆

The unit circle is the foundation for all trigonometric equations. A point on the circle at angle θ has coordinates (cos θ, sin θ), so the x-coordinate reads the cosine and the y-coordinate reads the sine.

单位圆是所有三角函数方程的基础。圆上角 θ 处的点坐标为 (cos θ, sin θ)，因此 x 坐标表示余弦，y 坐标表示正弦。

In quadrant I all three ratios are positive. In quadrant II only sin is positive, in quadrant III only tan is positive, and in quadrant IV only cos is positive. This sign pattern is often summarised as ‘CAST’ moving anticlockwise from the fourth quadrant.

在第一象限，三个比值均为正。第二象限只有 sin 为正，第三象限只有 tan 为正，第四象限只有 cos 为正。这个符号规律通常用 ‘CAST’ 来表示，从第四象限起按逆时针方向读取。

When solving equations, draw or imagine the CAST diagram to decide which quadrants the required angle lies in after taking the inverse trigonometric function.

解方程时，可画出或想象 CAST 图，在求反三角函数后判断所需角位于哪些象限。

2. Key Trigonometric Identities | 关键三角恒等式

Before rearranging an equation, you often need to rewrite it using identities. The most important identity is shown below.

在移项变形之前，你通常需要先用恒等式改写方程。最重要的恒等式如下所示。

sin² θ + cos² θ = 1

Two useful rearrangements are sin² θ = 1 – cos² θ and cos² θ = 1 – sin² θ. Also, dividing by cos² θ gives 1 + tan² θ = sec² θ, and dividing by sin² θ gives 1 + cot² θ = cosec² θ.

两个常用的变形为 sin² θ

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