θ/2 半角公式计算器（角度或已知 cos θ）Half-Angle Formula Calculator (from an Angle or a Known cos θ)

使用说明与常见问题How-to & FAQ

「半角公式计算器（角度或已知 cos θ）」是什么？What is the Half-Angle Formula Calculator (from an Angle or a Known cos θ) tool?

由角度 θ（度或弧度），或由已知 cos θ 的值和 θ/2 所在象限，用半角公式求 sin(θ/2)、cos(θ/2)、tan(θ/2)，正负号按象限确定，并列出公式和代入过程。本地处理。From an angle θ in degrees or radians, or from a known cos θ and the quadrant of θ/2, use the half-angle formulas to get sin(θ/2), cos(θ/2) and tan(θ/2), with signs set by the quadrant and the formulas and values shown. Processed locally.

如何使用「半角公式计算器（角度或已知 cos θ）」？How do I use the Half-Angle Formula Calculator (from an Angle or a Known cos θ) tool?

在「半角公式计算器（角度或已知 cos θ）」中，填写「已知」、「θ」、「单位」、「cos θ」、「θ/2 所在象限」等参数，结果会随输入实时计算并显示在下方。In the Half-Angle Formula Calculator (from an Angle or a Known cos θ): Fill in "Known", "θ", "Unit", "cos θ", "Quadrant of θ/2"; results are recalculated instantly as you type.

半角公式是什么？What are the half-angle formulas?

sin(θ/2) = ±√((1 − cos θ)/2)，cos(θ/2) = ±√((1 + cos θ)/2)，tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ)。sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2) and tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ).

正负号怎么确定？How is the sign chosen?

由 θ/2 所在的象限决定：第一象限都为正；第二象限 sin 正、cos 负；第三象限 sin、cos 都为负；第四象限 sin 负、cos 正。By the quadrant of θ/2: all positive in I; sin positive and cos negative in II; both negative in III; sin negative and cos positive in IV.

和二倍角公式有什么关系？How do they relate to the double-angle formulas?

把 cos 2α = 1 − 2sin²α = 2cos²α − 1 中的 α 换成 θ/2，再解出 sin(θ/2) 和 cos(θ/2) 就得到半角公式。Put α = θ/2 into cos 2α = 1 − 2sin²α = 2cos²α − 1 and solve for sin(θ/2) and cos(θ/2).

能举个计算实例吗？Can you show a worked example?

例如在「半角公式计算器（角度或已知 cos θ）」中填 已知 = 角度 θ，θ = 60，单位 = 度，结果区显示：cos θ：0.5；sin(θ/2) = ±√((1 − cos θ)/2)：0.5；cos(θ/2) = ±√((1 + cos θ)/2)：0.8660254038；tan(θ/2) = sin(θ/2)/cos(θ/2)：0.5773502692；θ/2：30°For example, entering Known = Angle θ, θ = 60, Unit = Degrees in the Half-Angle Formula Calculator (from an Angle or a Known cos θ) gives: cos θ: 0.5; sin(θ/2) = ±√((1 − cos θ)/2): 0.5; cos(θ/2) = ±√((1 + cos θ)/2): 0.8660254038; tan(θ/2) = sin(θ/2)/cos(θ/2): 0.5773502692; θ/2: 30°

哪些输入会被检查？Which inputs are validated?

「半角公式计算器（角度或已知 cos θ）」会检查这些输入——「θ」、「cos θ」：所选方式用到时必须填写；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Half-Angle Formula Calculator (from an Angle or a Known cos θ) checks its inputs — "θ", "cos θ": required when the chosen option uses it; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「半角公式计算器（角度或已知 cos θ）」的设置：「已知」可选 角度 θ、cos θ 的值（默认 角度 θ）；「θ」默认 60；「单位」可选 度、弧度（默认 度）；「cos θ」默认 0.5；「θ/2 所在象限」可选 第一象限、第二象限、第三象限、第四象限（默认 第一象限）。每次生成结果都按当前设置处理。Settings in the Half-Angle Formula Calculator (from an Angle or a Known cos θ) — "Known": Angle θ, Value of cos θ (default Angle θ); "θ" default 60; "Unit": Degrees, Radians (default Degrees); "cos θ" default 0.5; "Quadrant of θ/2": I, II, III, IV (default I). The result always uses the current settings.

计算精度如何？How precise are the results?

「半角公式计算器（角度或已知 cos θ）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Half-Angle Formula Calculator (from an Angle or a Known cos θ) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「半角公式计算器（角度或已知 cos θ）」免费吗？需要注册吗？Is the Half-Angle Formula Calculator (from an Angle or a Known cos θ) free? Do I need an account?

「半角公式计算器（角度或已知 cos θ）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Half-Angle Formula Calculator (from an Angle or a Known cos θ) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.