Trigonometric Functions: Complete A-Level Revision Guide | 三角函数：A-Level 完整复习指南

📚 Trigonometric Functions: Complete A-Level Revision Guide | 三角函数：A-Level 完整复习指南

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。

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

ference 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°。

dians 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 formulae only work with θ in radians. If θ is given in degrees, convert it first.

弧长与扇形面积公式只有在 θ 使用弧度时才成立。如果 θ 以角度给出，必须先转换。

s = rθ, A = ½r²θ

The exact values for common angles must be memorised. The table below gives sin θ, cos θ and tan θ for 0°, 30°, 45°, 60° and 90° in both degrees and radians.

常见角的精确值必须熟记。下表给出 0°、30°、45°、60°、90° 在角度与弧度下的 sin θ、cos θ 和 tan θ。

θ in degrees	0°	30°	45°	60°	90°
θ in radians	0	π / 6	π / 4	π / 3	π / 2
sin θ	0	1 / 2	√2 / 2	√3 / 2	1
cos θ	1	√3 / 2	√2 / 2	1 / 2	0
tan θ	0	1 / √3	1	√3	undefined

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2. Graphs and Periodicity | 图像与周期性

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。

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| shift is d. The same rules apply to cosine and tangent, except that tangent has period π / |b|.

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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，正切函数除外，其周期为 π / |b|。

Key graph features to remember: sin x is odd, cos x is even, and tan x has period π with asymptotes every π radians. Sketching these graphs helps you solve equations and inequalities accurately.

需要记住的关键图像特征：sin x 是奇函数，cos x 是偶函数，tan x 周期为 π，每隔 π 弧度有渐近线。画出这些图像有助于准确解方程和不等式。

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

defined as 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 / tan θ。

θ is undefined when sin θ = 0, and cot θ is undefined when sin θ = 0. The ranges are sec θ ∈ (−∞, −1] ∪ [1, ∞), cosec θ ∈ (−∞, −1] ∪ [1, ∞), and cot θ ∈ ℝ.

它们的定义域排除分母为零的值。当 cos θ = 0 时 sec θ 无定义，当 sin θ = 0 时 cosec θ 无定义，当 sin θ = 0 时 cot θ 无定义。值域为 sec θ ∈ (−∞, −1] ∪ [1, ∞)，cosec θ ∈ (−∞, −1] ∪ [1, ∞)，cot θ ∈ ℝ。

ple, y = sec x has asymptotes at x = π / 2 + nπ, and y = cosec x has asymptotes at x = nπ.

倒数函数的图像在原来函数为零的位置有竖直渐近线。例如，y = sec x 在 x = π / 2 + nπ 处有渐近线，y = cosec x 在 x = nπ 处有渐近线。

om sin²θ + cos²θ = 1 and are essential for simplifying expressions and proving identities.

涉及倒数函数的有用恒等式包括 1 + tan²θ = sec²θ 和 1 + cot²θ = cosec²θ。它们由 sin²θ + cos²θ = 1 推出，是化简表达式和证明恒等式的关键。

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4. Pythagorean and Quotient Identities | 勾股恒等式与商数恒等式

1. Dividing by cos²θ gives 1 + tan²θ = sec²θ. Dividing by sin²θ gives 1 + cot²θ = cosec²θ.

基本勾股恒等式为 sin²θ + cos²θ = 1。两边除以 cos²θ 得 1 + tan²θ = sec²θ。两边除以 sin²θ 得 1 + cot²θ = cosec²θ。

These are used constantly when simplifying trigonometric expressions or solving equations.

商数恒等式为 tan θ = sin θ / cos θ 和 cot θ = cos θ / sin θ。在化简三角表达式或解方程时经常使用。

n²θ + cos²θ = 1 gives cos²θ = 1 − (3 / 5)² = 16 / 25, so cos θ = 4 / 5 because θ is acute.

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