📚 Mastering Exact Trigonometric Values for 195° | 掌握 195° 的三角比值精确计算
In IGCSE Edexcel Mathematics, you are often required to work with trigonometric functions of angles beyond the familiar 0° to 90° range. The angle 195° serves as an excellent case study for applying the unit circle, reference angles, and exact value techniques. In this article, we will find sin 195°, cos 195°, and tan 195° without a calculator, expressing them in exact surd form and building a deeper understanding of trigonometric principles.
在 IGCSE Edexcel 数学中，你经常需要处理超出 0° 到 90° 范围的角的三角函数。195° 这个角是应用单位圆、参考角和精确值技巧的绝佳案例。在本文中，我们将在不使用计算器的情况下求出 sin 195°、cos 195° 和 tan 195°，并用根式的精确形式表示它们，从而加深对三角函数原理的理解。
1. Why Study Angles Beyond 90°? | 为什么学习大于 90° 的角？
Basic right‑triangle trigonometry restricts angles to acute measures, but the IGCSE syllabus broadens the definitions using the unit circle. This extension allows us to model periodic phenomena such as waves, oscillations, and circular motion. Angles like 195° appear naturally in bearings, in solving trigonometric equations, and when evaluating expressions that involve reflection and symmetry. Mastering these wider angles is essential for achieving top grades.
基本的直角三角形三角学将角限制在锐角范围内，但 IGCSE 大纲通过单位圆扩展了这些定义。这种扩展使我们能够对波动、振荡和圆周运动等周期性现象进行建模。像 195° 这样的角自然地出现在方位角、解三角方程以及涉及反射和对称性的求值问题中。掌握这些更广的角度对于取得高分至关重要。
2. Visualising 195° on the Unit Circle | 在单位圆上呈现 195°
The unit circle is a circle of radius 1 centred at the origin (0,0). An angle is measured anticlockwise from the positive x‑axis. Rotating 195° anticlockwise places the terminal side deep inside the third quadrant. At the intersection of this terminal side with the circle, the x‑coordinate equals cos 195° and the y‑coordinate equals sin 195°. Because both coordinates in the third quadrant are negative, we immediately know that cos 195° < 0 and sin 195° < 0.
单位圆是以原点 (0,0) 为圆心、半径为 1 的圆。角是从正 x 轴沿逆时针方向度量的。逆时针旋转 195° 会使终边深入到第三象限。在终边与圆的交点处，x 坐标等于 cos 195°，y 坐标等于 sin 195°。由于第三象限的两个坐标均为负值，我们立刻知道 cos 195° < 0 且 sin 195° < 0。
3. Reference Angle and the CAST Diagram | 参考角与 CAST 图
To connect an unfamiliar angle to a familiar acute one, we use the reference angle – the acute angle the terminal side makes with the x‑ ST diagram tells us which ratios are positive in each quadrant. In the third quadrant (the 'T' quadrant), only Tangent is positive, while Sine and Cosine are negative.
为了将陌生角与熟悉的锐角联系起来，我们使用参考角——终边与 x 轴所夹的锐角。对于第三象限的角，参考角就是 (角度 − 180°)。因此，对于 195°，参考角为 195° − 180° = 15°。CAST 图告诉我们每个象限中哪些比率为正。在第三象限（'T' 象限），只有正切为正，而正弦和余弦均为负。
Consequently, we can write:
因此，我们可以写出：
sin 195° = − sin 15° cos 195° = − cos 15° tan 195° = + tan 15°
This simple step transforms the problem into finding the exact values of 15°, a keystone angle that can be derived elegantly.
这个简单的步骤将问题转化为求 15° 的精确值，而 15° 是一个可以通过巧妙方式推导出来的基础角。
4. The Importance of 15° in Exact Value Work | 15° 在精确值计算中的重要性
In many IGCSE questions, you are either given the exact values of sin 15° and cos 15° or expected to know how to obtain them. Although 15° is not one of the standard 0°, 30°, 45°, 60°, 90° angles, it appears frequently because it can be built from these special angles. You might also encounter it when the question asks for values like sin 195°, sin 345°, or cos 165°.
在许多 IGCSE 题目中，要么会给出 sin 15° 和 cos 15° 的精确值，要么期望你知道如何得到它们。虽然 15° 不是标准的 0°、30°、45°、60°、90° 角之一，但由于它可以由这些特殊角构建，因此频繁出现。当题目要求计算 sin 195°、sin 345° 或 cos 165° 等值时，你也会遇到它。
5. Deriving Exact Values of sin 15° and cos 15° | 推导 sin 15° 和 cos 15° 的精确值
Method 1 – Using the difference formula (extension). Express 15° as 45° − 30° and apply the sine and cosine subtraction identities.
方法一——使用差角公式（拓展内容）。将 15° 表示为 45° − 30°，然后应用正弦和余弦的减法恒等式。
sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°
cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°
Substituting the known exact values sin 45° = √2/2, cos 30° = √3/2, cos 45° = √2/2, sin 30° = 1/2 yields:
代入已知的精确值 sin 45° = √2/2、cos 30° = √3/2、cos 45° = √2/2、sin 30° = 1/2 得到：
sin 15° = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4
cos 15° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
Method 2 – Half‑angle or geometric reasoning. The half‑ /2] gives the same result. Geometrically, you can start with an equilateral triangle to obtain a 30° angle, combine it with a right isosceles triangle, and construct a 15° angle to measure the ratios. Both approaches reinforce the surd forms above.
方法二——半角公式或几何推理。半角公式 sin 15° = √[(1 − cos 30°)/2] 给出相同结果。在几何上，你可以从一个等边三角形得到 30° 角，再与等腰直角三角形组合，构造出 15° 角来测量比值。两种方法都印证了上述根式形式。
For completeness, we also find tan 15° by division and rationalising:
为了完整性，我们也通过相除并有理化分母来求 tan 15°：
tan 15° = sin 15° / cos 15° = (√6 − √2) / (√6 + √2) = 2 − √3
(Rationalisation: multiply numerator and denominator by (√6 − √2), simplify the difference of squares, and reduce to 2 − √3.)
（有理化过程：分子分母同乘 (√6 − √2)，利用平方差化简，最后约分得到 2 − √3。）
6. Determining sin 195°, cos 195°, and tan 195° | 确定 sin 195°、cos 195° 和 tan 195°
Now return to 195° and apply the signs we established from the third quadrant. Remember that sin 195° and cos 195° are negative, while tan 195° is positive.
现在回到 195°，应用我们在第三象限确定的符号。记住 sin 195° 和 cos 195° 为负，而 tan 195° 为正。
sin 195° = − sin 15° = − (√6 − √2)/4 = (√2 − √6)/4
cos 195° = − cos 15° = − (√6 + √2)/4
tan 195° = + tan 15° = 2 − √3
The expression for sin 195° is often written as (√2 − √6)/4 to start with the positive term, though algebraically it is equivalent to −(√6 − √2)/4
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