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    "name": "Manazoo",
    "url": "https://manazoo.com/"
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  "license": "All rights reserved（AI の参照・要約と出典リンクつきの引用は可。転載・再配布・販売は不可）",
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  "disclaimer": "単元の並びは学習指導要領に沿って作り、年号や人物などの事実は公開データと照らし合わせています。誤りに気づいたら意見箱からお知らせください。確かめて直します。テストや受験の前には、教科書や先生の説明もあわせて確認してください。",
  "attribution": "事実データの一部は Wikidata（CC0）による。単元の構成は文部科学省「小学校学習指導要領」「中学校学習指導要領」「高等学校学習指導要領」（政府標準利用規約2.0）。",
  "kind": "entity",
  "slug": "trigonometric-function-derivatives",
  "name": "三角関数の導関数",
  "entity_kind": "concept",
  "wikidata_qid": "Q3296857",
  "wikidata_url": "https://www.wikidata.org/wiki/Q3296857",
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  "course": {
    "slug": "math-3",
    "title": "数学Ⅲ",
    "url": "https://manazoo.com/high/math-3/"
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  "unit": {
    "slug": "differential-calculus",
    "title": "微分法",
    "url": "https://manazoo.com/high/math-3/differential-calculus/"
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  "period": null,
  "place": null,
  "summary_points": [
    "sin xの導関数はcos x、cos xの導関数は−sin x、tan xの導関数は1/cos²x。",
    "これらは加法定理と、hを0に近づけるsin h/hと(1−cos h)/hの極限の式から導かれる。",
    "cos x＝sin(x＋π/2)、tan x＝sin x/cos xを使うと、cos xとtan xの導関数はsin xの導関数から導ける。"
  ],
  "facts": null,
  "key_points": [
    {
      "term": "(sin x)′",
      "note": "cos x"
    },
    {
      "term": "(cos x)′",
      "note": "−sin x"
    },
    {
      "term": "(tan x)′",
      "note": "1/cos²x"
    }
  ],
  "summary": null,
  "causality": null,
  "timeline": [],
  "related": {
    "cause": [],
    "result": [],
    "people": [],
    "same_period_japan": []
  },
  "questions": [
    {
      "id": "q-tfd-linear-combo-value-at-0",
      "type": "input",
      "prompt": "y＝3sin x−2cos xの導関数dy/dxを求め、x＝0でのdy/dxの値を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "dy/dx＝3cos x＋2sin xなので、x＝0ではcos0＝1、sin0＝0より、dy/dx＝3×1＋2×0＝3。",
      "entities": [
        "trigonometric-function-derivatives"
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    },
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      "id": "q-tfd-tangent-value-at-pi4",
      "type": "input",
      "prompt": "f(x)＝tan xの導関数f′(x)＝1/cos²xを使って、f′(π/4)の値を求めよ。",
      "choices": null,
      "answer": "2",
      "accept": [],
      "explanation": "cos(π/4)＝1/√2なので、cos²(π/4)＝1/2。f′(π/4)＝1/(1/2)＝2。",
      "entities": [
        "trigonometric-function-derivatives"
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      "verified": true,
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    },
    {
      "id": "q-tfd-cosine-derivative-choice",
      "type": "choice",
      "prompt": "cos xの導関数(cos x)′はどれか。",
      "choices": [
        "cos x",
        "−cos x",
        "−sin x",
        "sin x"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "cos x＝sin(x＋π/2)であることと(sin x)′＝cos xから、(cos x)′＝−sin xになる。",
      "entities": [
        "trigonometric-function-derivatives"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tfd-product-rule-choice",
      "type": "choice",
      "prompt": "f(x)＝sin x cos xの導関数f′(x)を積の導関数の公式で求めるとどれか。",
      "choices": [
        "cos2x",
        "2sin x cos x",
        "−sin2x",
        "sin2x"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "f′(x)＝(sin x)′cos x＋sin x(cos x)′＝cos²x−sin²x。これは2倍角の公式でcos2xと等しい。",
      "entities": [
        "trigonometric-function-derivatives"
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  "sources": [
    {
      "label": "文部科学省「高等学校学習指導要領（平成30年告示）解説 数学編 理数編」（数学Ⅲ (2)微分法 ア(ｳ)イ(ｱ) 三角関数の導関数について理解し求めること・「sin xの導関数は…加法定理と次の三角関数に関する極限の式limsinx/x=1、lim(1-cosx)/x=0を利用して導くことができる」。解説 p.75〜77）",
      "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf"
    }
  ],
  "media": [
    {
      "src": "/media/trigonometric-function-derivatives/sine-derivative-proof.png",
      "alt": "y＝sin xのグラフ上の点での接線の傾きがcosxに近づく様子と、加法定理で分けた極限の式を示した図",
      "license": "オリジナル（コードで作図）",
      "author": "Asatte Studio（Manazoo）",
      "source_url": "https://manazoo.com/credits/",
      "section": "why",
      "caption": "(sin x)′＝cos xの証明：加法定理で2つの項に分ける"
    },
    {
      "src": "/media/trigonometric-function-derivatives/sine-cosine-derivative-graphs.png",
      "alt": "y＝sin xのグラフとその導関数y＝cos xのグラフを上下に並べ、sin xの傾きが0になる場所でcos xが0を通ることを示した図",
      "license": "オリジナル（コードで作図）",
      "author": "Asatte Studio（Manazoo）",
      "source_url": "https://manazoo.com/credits/",
      "section": "formula",
      "caption": "sin xの傾き（導関数）がcos xになる様子"
    }
  ],
  "status": "published",
  "placeholder": false,
  "updated_at": "2026-10-31"
}