📚 Trigonometric Ratios of 135° | 135°角的三角函数
In IGCSE Edexcel Mathematics, extending trigonometric ratios beyond acute angles is a key skill. The angle 135° appears frequently in exact value problems, trigonometric equations, and coordinate geometry. This article thoroughly explains how to obtain and use the sine, cosine, and tangent of 135°, linking the concepts of reference angles, the unit circle, and the CAST diagram.
在IGCSE爱德思数学中，将三角函数比推广到锐角之外是一项关键技能。135°角经常出现在精确值问题、三角方程和坐标几何中。本文全面解释如何获得并使用135°的正弦、余弦和正切值，并将参考角、单位圆和CAST图联系起来。
1. Understanding Obtuse Angles and 135° | 理解钝角与135°
An angle between 90° and 180° is called an obtuse angle. 135° is a typical obtuse angle because it lies exactly halfway between 90° and 180°.
介于90°和180°之间的角称为钝角。135°是一个典型的钝角，因为它正好位于90°和180°的中间。
In many IGCSE problems, you cannot simply read its trigonometric values from a calculator set to degrees unless you are familiar with exact surd forms. Mastering 135° helps you handle all angles in the second quadrant.
在许多IGCSE问题中，除非你熟悉精确的根式形式，否则不能仅仅依靠设置为度数的计算器读取其三角函数值。掌握135°有助于你处理第二象限中的所有角度。
2. Positioning 135° on the Unit Circle | 在单位圆上定位135°
The unit circle has radius 1 centred at the origin. An angle of 135° is measured anticlockwise from the positive x‑axis. It terminates in the second quadrant, making an angle of 45° with the negative x‑axis.
单位圆是以原点为圆心、半径为1的圆。135°角从正x轴逆时针测量。其终边落在第二象限，与负x轴形成45°的角。
The coordinates of the point where the terminal side meets the unit circle are (-√2/2, √2/2). This directly gives cos 135° = -√2/2 and sin 135° = √2/2.
终边与单位圆的交点坐标为(-√2/2, √2/2)。这直接给出cos 135° = -√2/2 和 sin 135° = √2/2。
3. Finding the Reference Angle for 135° | 求135°的参考角
A reference angle is the acute angle the terminal side makes with the horizontal axis. For any angle θ in the second quadrant, the reference angle is 180° – θ.
参考角是终边与水平轴形成的锐角。对于第二象限中的任意角θ，参考角为180° – θ。
Thus, for 135° the reference angle is 180° – ls us that the magnitude of sin 135° and cos 135° will be the same as sin 45° and cos 45°.
因此，对于135°，参考角为180° – 135° = 45°。这告诉我们sin 135°和cos 135°的大小与sin 45°和cos 45°相同。
Using a reference angle simplifies finding exact trigonometric values, because you only need to adjust the sign according to the quadrant.
使用参考角可以简化精确三角值的求解，因为只需要根据象限调整符号。
4. Sine of 135° | 135°的正弦值
Using the symmetry property sin(180° – θ) = sin θ, we substitute θ = 45° to obtain sin 135° = sin 45°.
利用对称性 sin(180° – θ) = sin θ，代入θ = 45°，得到 sin 135° = sin 45°。
The exact value of sin 45° is √2 / 2 (or 1/√2 when rationalised differently). Therefore, sin 135° = √2 / 2.
sin 45°的精确值是√2 / 2（有理化后也可写作1/√2）。因此 sin 135° = √2 / 2。
Because sine corresponds to the y‑coordinate on the unit circle and is positive in the second quadrant, the positive value makes sense.
由于正弦对应单位圆上的y坐标，并且在第二象限为正，正值是合理的。
5. Cosine of 135° | 135°的余弦值
For cosine in the second quadrant, the identity cos(180° – θ) = –cos θ applies. With θ = 45°, we get cos 135° = –cos 45°.
对于第二象限中的余弦，恒等式 cos(180° – θ) = –cos θ 成立。代入θ = 45°，得到 cos 135° = –cos 45°。
Since cos 45° = √2 / 2, it follows that cos 135° = –√2 / 2.
因为 cos 45° = √2 / 2，所以 cos 135° = –√2 / 2。
The negative sign reflects that the x‑coordinate of the point on the unit circle is negative in the second quadrant.
负号反映了单位圆上点的x坐标在第二象限为负。
6. Tangent of 135° | 135°的正切值
Tangent is defined as sin θ ÷ cos θ. Using the values above, tan 135° = (√2/2) ÷ (–√2/2) = –1.
正切定义为 sin θ ÷ cos θ。利用上述值，tan 135° = (√2/2) ÷ (–√2/2) = –1。
Alternatively, apply the identity tan(180° – θ) = –tan θ. With θ = 45° and tan 45° = 1, we immediately get tan 135° = –1.
或者应用恒等式 tan(180° – θ) = –tan θ。由θ = 45°且 tan 45° = 1，立刻得到 tan 135° = –1。
Tangent being negative in the second quadrant is consistent with the CAST diagram rule where only Sine is positive.
正切在第二象限为负，与CAST图中只有正弦为正的规则一致。
7. Using the CAST Diagram to Determine Signs | 利用CAST图确定符号
The CAST diagram divides the coordinate plane into quadrants, labelling which ratios are positive:
CAST图将坐标平面分为象限，并标明哪些比为正：
- Quadrant I (0°–90°): All (Cos, Sin, Tan positive) | 第一象限 (0°–90°): 全部为正
- Quadrant II (90°–180°): Sin positive, others negative | 第二象限 (90°–180°): 正弦为正，其余为负
- Quadrant III (180°–270°): Tan positive | 第三象限 (180°–270°): 正切为正
- Quadrant IV (270°–360°): Cos positive | 第四象限 (270°–360°): 余弦为正
°, located in Quadrant II, sin is positive, cos is negative, so tan = sin/cos is negative.
对于位于第二象限的135°，sin为正，cos为负，因此tan = sin/cos 为负。
Many students find the CAST diagram memorable because the word CAST reads from the fourth quadrant back to the first: C (cos), A (all), S (sin), T (tan).
许多学生发现CAST图好记，因为单词CAST从第四象限倒数回第一象限：C (cos), A (all), S (sin), T (tan)。
8. Exact Values from the 45°‑45°‑90° Triangle | 来自45°‑45°‑90°三角形的精确值
A right‑angled isosceles triangle with angles 45°, 45°, 90° has side lengths in the ratio 1 : 1 : √2.
一个角为45°、45°、90°的等腰直角三角形，其边长比例为1 : 1 : √2。
From this triangle, sin 45° = opposite/hypotenuse = 1/√2 = √2/2, cos 45° = adjacent/hypotenuse = 1/√2 = √2/2, tan 45° = 1.
由此三角形，sin 45° = 对边/斜边 = 1/√2 = √2/2，cos 45° = 邻边/斜边 = 1/√2 = √2/2，tan 45° = 1。
These acute values are then used with the quadrant sign adjustments to write the exact values for 135°.
然后利用这些锐角值，经过象限符号调整，写出135°的精确值。
Hence, sin 135° = √2/2, cos 135° = –√2/2, tan 135° = –1.
因此，sin 135° = √2/2, cos 135° = –√2/2, tan 135° = –1。
9. Verifying with Trigonometric Graphs | 用三角函数图像验证
o a maximum of 1 at 90° and then falls symmetrically. At x = 135°, the height is sin 45° = √2/2.
y = sin x 在0°到180°之间的图像在90°时上升到最大值1，然后对称下降。在x = 135°时，高度为 sin 45° = √2/2。
The graph of y = cos x starts at 1, crosses zero at 90°, and reaches –1 at 180°. At 135°, the value is –cos 45° = –√2/2.
y = cos x 的图像从1开始，在90°时穿过零，在180°时达到–1。在135°时，值为 –cos 45° = –√2/2。
These graphical features reassure us that the exact surd values align with the wave shapes of sine and cosine.
这些图像特征保证了精确根式值与正弦和余弦的波形形状一致。
10. Worked Examples Involving 135° | 涉及135°的典型例题
Example 1: Evaluate sin 135° + cos 135°.
Solution: sin 135° + cos 135° = √2/2 + (–√2/2) = 0.
例题1：计算 sin 135° + cos 135°。
解： sin 135° + cos 135° = √2/2 + (–√2/2) = 0。
Example 2: Solve sin x = √2/2 for 0° ≤ x ≤ 360°.
he acute solution is x = 45°. Since sine is also positive in the second quadrant, x = 180° – 45° = 135° is the other solution. Answer: x = 45°, 135°.
例题2：在0° ≤ x ≤ 360°内解方程 sin x = √2/2。
锐角解为 x = 45°。由于正弦在第二象限也为正，另一解为 x = 180° – 45° = 135°。答案：x = 45°, 135°。
Example 3: Given that tan θ = –1 and θ is obtuse, find θ.
Obtuse means between 90° and 180°. The reference angle for tan = 1 is 45°, so θ = 180° – 45° = 135°.
例题3：已知 tan θ = –1 且 θ 为钝角，求 θ。
钝角意味着在90°与180°之间。tan = 1 的参考角是45°，所以 θ = 180° – 45° = 135°。
11. Common Mistakes and How to Avoid Them | 常见错误与如何避免
Mistake 1: Writing sin 135° = –√2/2. Remember sine is positive in the second quadrant; only the cosine and tangent become negative.
错误1：写作 sin 135° = –√2/2。记住正弦在第二象限为正；只有余弦和正切变为负。
Mistake 2: Using the wrong reference angle, e.g. 135° – 90° = 45° is not the reference; the reference is 180° – Although numerically identical here, the correct definition matters for angles like 100°.
错误2：使用错误的参考角，例如135° – 90° = 45°并非参考角；参考角应为180° – 135° = 45°。尽管这里数值相同，但对于100°等角，准确定义很重要。
3: Forgetting to negative the cosine when using symmetric identities. The formula cos(180° – θ) = –cos θ must include the minus sign.
错误3：使用对称恒等式时忘记给余弦加负号。公式 cos(180° – θ) = –cos θ 必须包含负号。
Always check with the CAST diagram or draw a quick sketch of the angle to confirm signs.
经常用CAST图检查，或快速画出角度草图来确认符号。
12. Summary Table and Key Takeaways | 总结表格与核心要点
| Angle (θ) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 45° | √2/2 | √2/2 | 1 |
| 135° | √2/2 | -√2/2 | -1 |
| 225° | -√2/2 | -√2/2 | 1 |
| 315° | -√2/2 | √2/2 | -1 |
The exact values for 135° are sin 135° = √2/2, cos 135° = –√2/2, tan 135° = – foundation for solving equations, working with bearings, and modelling periodic behaviour.
135°的精确值为 sin 135° = √2/2, cos 135° = –√2/2, tan 135° = –1。这些是解方程、处理方位角和建模周期现象的基础。
Remembering that the reference angle is 45° and that only sine is positive in the second quadrant will allow you to re‑derive these values confidently in any exam.
记住参考角是45°且第二象限只有正弦为正，将让你在任何考试中都能自信地重新推导这些值。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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