Trigonometric Functions | 三角函数

📚 Trigonometric Functions | 三角函数

Trigonometric functions describe the relationship between angles and side ratios in right-angled triangles, and they extend far beyond geometry into waves, oscillations, calculus and modelling. In Edexcel A Level Mathematics, you must be confident with radians, exact values, graphs, identities, equations, reciprocal functions, harmonic form and calculus applied to trigonometric functions.

三角函数描述直角三角形中角与边比之间的关系，并远不止应用于几何，还广泛用于波动、振荡、微积分与建模。在 Edexcel A Level 数学中，你必须熟练掌握弧度制、精确值、图像、恒等式、方程、倒数函数、谐波形式以及三角函数的微积分。

1. Radian Measure and Exact Values | 弧度制与精确值

A radian measures an angle by the ratio of arc length to radius: θ = s/r. Since the circumference of a full circle is 2πr, one complete turn is 2π radians. Therefore, π rad = 180°.

弧度制用弧长与半径之比度量角：θ = s/r。整圆的周长为 2πr，所以一整圈为 2π 弧度，因此 π rad = 180°。

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 60° = π/3 rad and 150° = 5π/6 rad.

将角度转换为弧度，乘以 π/180；将弧度转换为角度，乘以 180/π。例如，60° = π/3 rad，150° = 5π/6 rad。

Arc length and sector area only work with θ in radians:

弧长与扇形面积公式只有在 θ 使用弧度时才成立：

s = rθ, A = ½r²θ

The exact values for common angles must be memorised:

常见角的精确值必须熟记：

2. Graphs and Periodicity | 图像与周期性

The sine and cosine functions have period 2π and range [−1, 1]. The tangent function has period π and is undefined at x = π/2 + nπ, where its graph has vertical asymptotes.

正弦与余弦函数的周期为 2π，值域为 [−1, 1]。正切函数的周期为 π，在 x = π/2 + nπ 处无定义，图像在这些位置有竖直渐近线。

The graph of y = sin x passes through the origin, reaches its maximum at x = π/2 and returns to zero at x = π. The graph of y = cos x starts at its maximum at x = 0 and has the same shape shifted left by π/2.

y = sin x 的图像经过原点，在 x = π/2 处达到最大值，并在 x = π 处回到零。y = cos x 的图像在 x = 0 处从最大值开始，形状与正弦相同但向左平移 π/2。

For a transformed function y = a sin(bx + c) + d, the amplitude is |a|, the period is 2π/|b|, the phase shift is −c/b and the vertical shift is d. The same rules apply to cosine.

对于变换后的函数 y = a sin(bx + c) + d，振幅为 |a|，周期为 2π/|b|，相位移为 −c/b，纵向平移为 d。余弦函数也遵循相同规则。

3. Reciprocal Functions sec, cosec, cot | 倒数三角函数

The reciprocal trigonometric functions are sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = cos θ/sin θ = 1/tan θ.

倒数三角函数定义为 sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = cos θ/sin θ = 1/t

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