{
  "api_version": "v1",
  "publisher": {
    "name": "Manazoo",
    "url": "https://manazoo.com/"
  },
  "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）",
  "terms_url": "https://manazoo.com/terms/",
  "disclaimer": "内容の正確さには気をつけていますが、すべてが正しいとは限りません。テストや受験の前には、教科書や先生の説明でも確かめてください。間違いを見つけたら意見箱から知らせてください。",
  "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「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": "quiz",
  "unit": {
    "slug": "trigonometric-functions",
    "title": "三角関数",
    "url": "https://manazoo.com/high/math-2/trigonometric-functions/"
  },
  "canonical_url": "https://manazoo.com/high/math-2/trigonometric-functions/quiz/",
  "questions": [
    {
      "id": "q-dafc-composition-max-value",
      "type": "input",
      "prompt": "4sinθ＋3cosθの最大値を求めよ。",
      "choices": null,
      "answer": "5",
      "accept": [],
      "explanation": "合成するとR＝√(4²＋3²)＝√25＝5なので、4sinθ＋3cosθ＝5sin(θ＋α)の形になり、最大値はR＝5。",
      "entities": [
        "double-angle-formula-and-composition"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dafc-cos2alpha-choice",
      "type": "choice",
      "prompt": "cosα＝1/3のとき、cos2αの値はどれか。",
      "choices": [
        "−1/3",
        "7/9",
        "−7/9",
        "2/9"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "cos2α＝2cos²α－1＝2×(1/9)－1＝2/9－1＝−7/9。4つ目は「－1」を忘れて2cos²αだけで止めた誤り。",
      "entities": [
        "double-angle-formula-and-composition"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dafc-sin2alpha-value",
      "type": "input",
      "prompt": "sinα＝3/5、cosα＝4/5のとき、sin2αの値を求めよ。",
      "choices": null,
      "answer": "24/25",
      "accept": [
        "0.96"
      ],
      "explanation": "sin2α＝2sinαcosα＝2×(3/5)×(4/5)＝24/25。",
      "entities": [
        "double-angle-formula-and-composition"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dafc-tan2alpha-choice",
      "type": "choice",
      "prompt": "tanα＝1/2のとき、tan2αの値はどれか。",
      "choices": [
        "1",
        "2",
        "3/4",
        "4/3"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "tan2α＝2tanα／(1－tan²α)＝(2×1/2)／(1－1/4)＝1／(3/4)＝4/3。",
      "entities": [
        "double-angle-formula-and-composition"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-garm-degree-to-radian-315",
      "type": "input",
      "prompt": "315°を弧度法で表すと、(□/4)πラジアンになる。□に入る数を答えよう。",
      "choices": null,
      "answer": "7",
      "accept": [],
      "explanation": "315°をラジアンに直すには180で割ってπをかける。315÷180＝7/4なので、315°＝7/4πラジアン。分母が4のとき、分子は7になる。",
      "entities": [
        "general-angles-and-radian-measure"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-garm-general-angle-840",
      "type": "choice",
      "prompt": "840°を、θ＋360°×n（0°≤θ＜360°、nは整数）の形で表すと、どれになるか。",
      "choices": [
        "60°＋360°×2",
        "120°＋360°×2",
        "120°＋360°×3",
        "240°＋360°×1"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "840÷360＝2あまり120なので、840°＝120°＋360°×2。2つ目は1200°、3つ目は600°、4つ目は780°になり、どれも840°と一致しない。",
      "entities": [
        "general-angles-and-radian-measure"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-garm-sector-arc-r8",
      "type": "input",
      "prompt": "半径8、中心角(3/4)πラジアンの扇形の弧の長さは、□πになる。□に入る数を答えよう。",
      "choices": null,
      "answer": "6",
      "accept": [],
      "explanation": "弧の長さはl＝rθに代入して、l＝8×3/4π＝6π。□に入る数は6。",
      "entities": [
        "general-angles-and-radian-measure"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-garm-sector-area-r8",
      "type": "input",
      "prompt": "同じ扇形（半径8、中心角(3/4)πラジアン）の面積は、□πになる。□に入る数を答えよう。",
      "choices": null,
      "answer": "24",
      "accept": [],
      "explanation": "面積はS＝(1/2)r²θに代入して、S＝(1/2)×8²×3/4π＝24π。□に入る数は24。",
      "entities": [
        "general-angles-and-radian-measure"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-gtf-amplitude-3sintheta",
      "type": "choice",
      "prompt": "y＝3sinθのグラフの最大値はいくつか。",
      "choices": [
        "1/3",
        "1",
        "2",
        "3"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "sinθの最大値は1なので、y＝3sinθの最大値は3×1＝3になる。",
      "entities": [
        "graphs-of-trigonometric-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-gtf-period-cos3theta",
      "type": "choice",
      "prompt": "y＝cos3θの周期はどれか。",
      "choices": [
        "2π",
        "3π",
        "2/3π",
        "π/3"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "角についた係数3はθの進み方を3倍速くするので、周期は2πの1/3の2/3πになる。3πは3をかけてしまった誤り、2πは係数を無視した誤り。",
      "entities": [
        "graphs-of-trigonometric-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-gtf-period-phase-shift",
      "type": "choice",
      "prompt": "y＝sin(θ−π/3)の周期はどれか。",
      "choices": [
        "2π",
        "π",
        "π/3",
        "4π"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "θ−π/3はθ軸方向にπ/3だけ平行移動しただけで、角についた係数は1のまま変わらない。周期はy＝sinθと同じ2πになる。",
      "entities": [
        "graphs-of-trigonometric-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-gtf-periodic-phenomena-choice",
      "type": "choice",
      "prompt": "観覧車の高さのように、一定時間ごとに同じ値を繰り返す現象を表すのに適した関数はどれか。",
      "choices": [
        "一次関数",
        "二次関数",
        "三角関数",
        "指数関数"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "一次関数・二次関数・指数関数は値が一方向に増え続けたり変わり続けたりする関数で、同じ値を繰り返さない。周期的に繰り返す現象は三角関数で表せる。",
      "entities": [
        "graphs-of-trigonometric-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tat-cos-addition-value",
      "type": "input",
      "prompt": "sinα＝3/5、cosα＝4/5、β＝π（sinβ＝0、cosβ＝−1）のとき、cos(α＋β)の値を求めよ。",
      "choices": null,
      "answer": "-4/5",
      "accept": [
        "−4/5",
        "－4/5"
      ],
      "explanation": "cos(α＋β)＝cosαcosβ－sinαsinβ＝(4/5)×(−1)－(3/5)×0＝−4/5。",
      "entities": [
        "trigonometric-addition-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tat-sin-addition-formula-choice",
      "type": "choice",
      "prompt": "sin(α＋β)の加法定理として正しい式はどれか。",
      "choices": [
        "sinαsinβ＋cosαcosβ",
        "sinαcosβ＋cosαsinβ",
        "sinαcosβ－cosαsinβ",
        "cosαcosβ－sinαsinβ"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "sin(α＋β)＝sinαcosβ＋cosαsinβが正しい。2つ目はsin(α－β)、3つ目はcos(α＋β)、4つ目はcos(α－β)の式になっている。",
      "entities": [
        "trigonometric-addition-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tat-sin-addition-value",
      "type": "input",
      "prompt": "sinα＝3/5、cosα＝4/5、β＝π/2（sinβ＝1、cosβ＝0）のとき、sin(α＋β)の値を求めよ。",
      "choices": null,
      "answer": "4/5",
      "accept": [
        "0.8"
      ],
      "explanation": "sin(α＋β)＝sinαcosβ＋cosαsinβ＝(3/5)×0＋(4/5)×1＝4/5。",
      "entities": [
        "trigonometric-addition-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tat-tan-addition-choice",
      "type": "choice",
      "prompt": "tanα＝1、tanβ＝1/3のとき、tan(α＋β)の値はどれか。",
      "choices": [
        "1/2",
        "2",
        "3/2",
        "4/3"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "tan(α＋β)＝(tanα＋tanβ)／(1－tanαtanβ)＝(1＋1/3)／(1－1×1/3)＝(4/3)／(2/3)＝2。",
      "entities": [
        "trigonometric-addition-theorem"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tfga-cos-from-sin-second-quadrant",
      "type": "choice",
      "prompt": "sinθ＝3/5（θは第2象限の角）のとき、cosθの値はどれか。",
      "choices": [
        "−4/5",
        "−3/5",
        "4/5",
        "3/5"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "sin²θ＋cos²θ＝1より、cos²θ＝1－9/25＝16/25、cosθ＝±4/5。第2象限はcosθが負なので、cosθ＝−4/5。",
      "entities": [
        "trigonometric-functions-of-general-angles"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tfga-cos300",
      "type": "input",
      "prompt": "θ＝300°のとき、cosθの値を求めよ。",
      "choices": null,
      "answer": "1/2",
      "accept": [
        "0.5"
      ],
      "explanation": "300°＝360°－60°で、cos(−θ)＝cosθの関係からcos300°＝cos(−60°)＝cos60°＝1/2。",
      "entities": [
        "trigonometric-functions-of-general-angles"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tfga-sin-cos-tan-120",
      "type": "choice",
      "prompt": "θ＝120°のとき、sinθ、cosθ、tanθの値の組み合わせとして正しいものはどれか。",
      "choices": [
        "sinθ＝1/2、cosθ＝−√3/2、tanθ＝−√3/3",
        "sinθ＝−√3/2、cosθ＝−1/2、tanθ＝√3",
        "sinθ＝√3/2、cosθ＝−1/2、tanθ＝−√3",
        "sinθ＝√3/2、cosθ＝1/2、tanθ＝√3"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "120°＝180°－60°なので、単位円上の点は(−1/2, √3/2)。sin120°＝√3/2、cos120°＝−1/2、tan120°＝(√3/2)÷(−1/2)＝−√3。2つ目は150°、3つ目は60°、4つ目は240°の値になっている。",
      "entities": [
        "trigonometric-functions-of-general-angles"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tfga-sin135-choice",
      "type": "choice",
      "prompt": "θ＝135°のとき、sinθの値はどれか。",
      "choices": [
        "−√2/2",
        "√2/2",
        "−1/2",
        "1/2"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "135°＝180°－45°なので、単位円上の点は(−√2/2, √2/2)。sin135°＝√2/2。第2象限なのでsinθは正になる。",
      "entities": [
        "trigonometric-functions-of-general-angles"
      ],
      "verified": true,
      "placeholder": false
    }
  ]
}