2θ 二倍角公式计算器（角度或已知函数值）Double-Angle Formula Calculator (from an Angle or a Known Value)
由角度 θ（度或弧度），或由已知的 sin θ、cos θ 值和 θ 所在象限，计算 sin 2θ、cos 2θ、tan 2θ，并列出所用的二倍角公式及代入过程。本地处理。From an angle θ in degrees or radians, or from a known sin θ or cos θ and the quadrant of θ, get sin 2θ, cos 2θ and tan 2θ with the double-angle formulas used and the substituted values. Processed locally.
使用说明与常见问题How-to & FAQ
「二倍角公式计算器（角度或已知函数值）」是什么？What is the Double-Angle Formula Calculator (from an Angle or a Known Value) tool?
由角度 θ（度或弧度），或由已知的 sin θ、cos θ 值和 θ 所在象限，计算 sin 2θ、cos 2θ、tan 2θ，并列出所用的二倍角公式及代入过程。本地处理。From an angle θ in degrees or radians, or from a known sin θ or cos θ and the quadrant of θ, get sin 2θ, cos 2θ and tan 2θ with the double-angle formulas used and the substituted values. Processed locally.
如何使用「二倍角公式计算器（角度或已知函数值）」？How do I use the Double-Angle Formula Calculator (from an Angle or a Known Value) tool?
在「二倍角公式计算器（角度或已知函数值）」中，填写「已知」、「θ」、「单位」、「函数值」、「θ 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 Double-Angle Formula Calculator (from an Angle or a Known Value): Fill in "Known", "θ", "Unit", "Value", "Quadrant of θ"; results are recalculated instantly as you type.
二倍角公式有哪些？What are the double-angle formulas?
sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ；tan 2θ = 2tan θ ÷ (1 − tan²θ)。sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2tan θ / (1 − tan²θ).
只知道 sin θ 怎么办？What if I only know sin θ?
由 cos θ = ±√(1 − sin²θ) 求 cos θ，正负号由 θ 所在象限决定（第一、四象限 cos 为正）。选择象限后本页会自动处理。Use cos θ = ±√(1 − sin²θ), with the sign set by the quadrant of θ (cos is positive in quadrants I and IV). Choose the quadrant and the page handles it.
tan 2θ 什么时候无定义？When is tan 2θ undefined?
当 cos 2θ = 0，即 2θ = 90° + k·180°（θ = 45° + k·90°）时，tan 2θ 无定义。When cos 2θ = 0, that is 2θ = 90° + k·180° (θ = 45° + k·90°).
能举个计算实例吗？Can you show a worked example?
例如在「二倍角公式计算器（角度或已知函数值）」中填 已知 = 角度 θ，θ = 30，单位 = 度，结果区显示：sin θ, cos θ：0.5, 0.8660254038；sin 2θ = 2 sin θ cos θ：0.8660254038；cos 2θ = cos²θ − sin²θ：0.5；= 2cos²θ − 1 = 1 − 2sin²θ：0.5 = 0.5；tan 2θ = 2tan θ / (1 − tan²θ)：1.7320508076；2θ：60°For example, entering Known = Angle θ, θ = 30, Unit = Degrees in the Double-Angle Formula Calculator (from an Angle or a Known Value) gives: sin θ, cos θ: 0.5, 0.8660254038; sin 2θ = 2 sin θ cos θ: 0.8660254038; cos 2θ = cos²θ − sin²θ: 0.5; = 2cos²θ − 1 = 1 − 2sin²θ: 0.5 = 0.5; tan 2θ = 2tan θ / (1 − tan²θ): 1.7320508076; 2θ: 60°
哪些输入会被检查？Which inputs are validated?
「二倍角公式计算器（角度或已知函数值）」会检查这些输入——「θ」、「函数值」：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 Double-Angle Formula Calculator (from an Angle or a Known Value) checks its inputs — "θ", "Value": 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?
「二倍角公式计算器（角度或已知函数值）」的设置：「已知」可选 角度 θ、sin θ 的值、cos θ 的值（默认 角度 θ）；「θ」默认 30；「单位」可选 度、弧度（默认 度）；「函数值」默认 0.6；「θ 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 Double-Angle Formula Calculator (from an Angle or a Known Value) — "Known": Angle θ, Value of sin θ, Value of cos θ (default Angle θ); "θ" default 30; "Unit": Degrees, Radians (default Degrees); "Value" default 0.6; "Quadrant of θ": I, II, III, IV (default I). The result always uses the current settings.
计算精度如何？How precise are the results?
「二倍角公式计算器（角度或已知函数值）」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 Double-Angle Formula Calculator (from an Angle or a Known Value) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「二倍角公式计算器（角度或已知函数值）」免费吗？需要注册吗？Is the Double-Angle Formula Calculator (from an Angle or a Known Value) free? Do I need an account?
「二倍角公式计算器（角度或已知函数值）」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 Double-Angle Formula Calculator (from an Angle or a Known Value) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.