Simple Trigonometric Equations | 简单三角方程

📚 Simple Trigonometric Equations | 简单三角方程

ometric function of an unknown angle is set equal to a constant. Typical examples are sin θ = 0.5, cos 2θ = √3 nd every solution inside a given interval using the CAST diagram, exact values for 30°, 45° and 60°, and angle transformations, working confidently in both degrees and radians.

这篇 Edexcel A-level 数学复习指南讲解如何求解简单三角方程，也就是某个未知角的单一三角函数等于一个常数的方程。典型例子包括 sin θ = 0.5、cos 2θ = √3/2 和 tan θ = 1。你将学会如何在指定区间内求出全部解，所用工具包括 CAST 图、30°、45°、60° 的特殊角精确值以及角度变换，并能熟练地在角度制和弧度制之间切换。

1. What Are Simple Trigonometric Equations? | 什么是简单三角方程？

A simple trigonometric equation is an equation in which one trigonometric function of an unknown angle is equal to a constant. The unknown quantity is the angle itself, usually written θ or x. The equation is called 'simple' because only one trigonometric function appears, and its argument is a single angle rather than a sum or a multiple of angles. Solving the equation means finding every angle that makes the statement true.

简单三角方程是指未知角的某一个三角函数等于常数的方程，未知量是角本身，通常记作 θ 或 x。之所以称为"简单"，是因为方程中只出现一个三角函数，而且它的自变量是单个角，而不是角的和或倍数。求解方程就是要找出所有能使等式成立的角。

ngle is unrestricted. The sine, cosine and tangent functions are periodic, so the same value repeats again and again as the angle increases. Exam questions therefore restrict the work to a stated interval, such as 0° ≤ θ < 360° or 0 ≤ θ < 2π.

与 3x = 12 这类只有一个解的一次方程不同，当角不受限制时，三角方程通常有无穷多个解。正弦、余弦和正切函数都是周期函数，因此随着角度增大，同一个函数值会不断重复出现。所以考题通常会把范围限制在某个指定区间内，例如 0° ≤ θ < 360° 或 0 ≤ θ < 2π。

• One trigonometric function only, for example sin θ, cos 2θ or tan θ. | 只含一个三角函数，例如 sin θ、cos 2θ 或 tan θ。
• The right-hand side is a constant, such as 0.5, −√3/2 or 1. | 等式右边是一个常数，例如 0.5、−√3/2 或 1。
• The answer must be given in the stated interval, in the stated angle measure. | 答案必须写在题目给定的区间内，并使用题目要求的角的单位。

2. Why There Are Multiple Solutions | 为什么会有多个解

ity is the reason a single value of sin θ, cos θ or tan θ corresponds to more than one angle inside a 360° interval. On the graph of y = sin θ, the horizontal line y = 0.5 cuts the curve twice between 0° and 360°, at θ = 30° and θ = 150°.

正弦和余弦函数每 360°（即 2π 弧度）重复一次，正切函数每 180°（即 π 弧度）重复一次。正是这种周期性，使得同一个 sin θ、cos θ 或 tan θ 的值在 360° 区间内会对应不止一个角。在 y = sin θ 的图像上，水平直线 y = 0.5 在 0° 到 360° 之间与曲线相交两次，分别在 θ = 30° 和 θ = 150°。

Symmetry provides a quick way to find the partner solution. For sine, the second solution in the same cycle is 180° − θ₀. For cosine, it is 360° − θ₀, which is the same as −θ₀. For tangent, the next solution is simply θ₀ + 180°, because the tangent graph repeats twice as fast as the sine and cosine graphs.

对称性提供了快速找到另一个解的方法。对于正弦，同一周期内的第二个解是 180° − θ₀；对于余弦，第二个解是 360° − θ₀，也就是 −θ₀；对于正切，下一个解直接是 θ₀ + 180°，因为正切图像的重复速度是正弦和余弦图像的两倍。

sin θ: θ₀ and 180° − θ₀　|　cos θ: θ₀ and 360° − θ₀　|　tan θ: θ₀ + 180°n

3. The CAST Diagram and Quadrants | CAST 图与象限

The CAST diagram is a memory aid that shows which trigonometric functions are positive in each quadrant of the unit circle. Starting from the positive x-axis and moving anticlockwise, the four letters stand for All, Sin, Tan and Cos. In the first quadrant all three functions are positive; in the second only sine is positive; in the third only tangent is positive; in the fourth only cosine is positive.

CAST 图是一种记忆工具，用来表示单位圆各象限中哪些三角函数为正。从正 x 轴出发逆时针旋转，四个字母依次代表 All（全部）、Sin（正弦）、Tan（正切）、Cos（余弦）。第一象限三个函数都为正；第二象限只有正弦为正；第三象限只有正切为正；第四象限只有余弦为正。

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To use the diagram, first find the acute angle with the horizontal axis, usually called the principal value or the reference angle. Then decide which quadrants can contain a solution by checking the sign of the given constant. For example, if sin θ is negative, solutions must lie in the third and fourth quadrants, where sine is negative.

使用该图时，先求出与水平轴之间的锐角，通常称为主值或参考角；然后根据给定常数的正负号判断哪些象限可能含有解。例如，如果 sin θ 为负，解必须位于第三和第四象限，因为那里的正弦为负。

4. Exact Values You Must Know | 必须掌握的特殊角精确值

Edexcel questions frequently require exact answers rather than decimals, so you must know the sine, cosine and tangent of 0°, 30°, 45°, 60° and 90°. These values come from the two standard right-angled triangles: the half-square, which gives 45°, and the half-equilateral triangle, which gives 30° and 60°.

Edexcel 考题经常要求精确答案而不是小数，因此你必须掌握 0°、30°、45°、60° 和 90° 的正弦、余弦和正切值。这些值来自两个标准直角三角形：半正方形给出 45°，半等边三角形给出 30° 和 60°。

are 0, π/6, π/4, π/3 and π/2. Notice that sin 30° = cos 60° = 1/2 and sin 60° = cos 30° = √3 complementary functions. Also remember that tan θ = sin θ ÷ cos θ, so tan 30° = (1/2) ÷ (√3/2) = 1/√3 = √3/3.

用弧度表示时，这些角分别是 0、π/6、π/4、π/3 和 π/2。注意 sin 30° = cos 60° = 1/2，sin 60° = cos 30° = √3/2，因为正弦和余弦是互余函数。还要记住 tan θ = sin θ ÷ cos θ，所以 tan 30° = (1/2) ÷ (√3/2) = 1/√3 = √3/3。

5. Solving sin θ = k | 求解 sin θ = k

, making sure the calculator is in the correct mode. If k is positive, θ₀ lies in the first quadrant. If k is negative, the calculator returns a negative angle, so add 360° to place it in the fourth quadrant. The second solution in the same 360° cycle is always 180° − θ₀.

第一步是用计算器求出主值 θ₀ = sin⁻¹ k，并确保计算器处于正确的角度模式。如果 k 为正，θ₀ 在第一象限；如果 k 为负，计算器会返回一个负角，因此要加上 360° 把它放到第四象限。同一 360° 周期内的第二个解总是 180° − θ₀。

n θ = 0.5, so the answer is θ = 30° and θ = 150°. To solve sin θ = −0.5, the principal value from the calculator is −30°, so the two solutions in the interval are 210° and 330°.

例如，在区间 0° ≤ θ < 360° 内求解 sin θ = 0.5，主值为 θ₀ = 30°，第二个解为 180° − 30° = 150°。两个角都满足 sin θ = 0.5，所以答案是 θ = 30° 和 θ = 150°。若求解 sin θ = −0.5，计算器给出的主值是 −30°，因此区间内的两个解是 210° 和 330°。

sin θ = k → θ = θ₀ + 360°n or θ = 180° − θ₀ + 360°n

After finding the two basic solutions, keep adding or subtracting 360° until the values fall outside the required interval. If the equation contains sin 2θ or sin(θ + 30°), do not add 360° to θ directly; instead solve for the whole angle 2θ or θ + 30° over an extended interval, then convert back to θ.

找到两个基本解后，不断加或减 360°，直到数值超出题目要求的区间为止。如果方程中含有 sin 2θ 或 sin(θ + 30°)，不要直接给 θ 加 360°，而应先在整个角 2θ 或 θ + 30° 的扩展区间上求解，再换算回 θ。

6. Solving cos θ = k | 求解 cos θ = k

For cosine equations, the principal value

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