{
  "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": "figures-and-measurement",
    "title": "図形と計量",
    "url": "https://manazoo.com/high/math-1/figures-and-measurement/"
  },
  "canonical_url": "https://manazoo.com/high/math-1/figures-and-measurement/quiz/",
  "questions": [
    {
      "id": "q-loc-a-from-b5-c8-a60",
      "type": "input",
      "prompt": "三角形ABCで、b＝5、c＝8、A＝60°のとき、辺aの長さを求めよ。",
      "choices": null,
      "answer": "7",
      "accept": [],
      "explanation": "a²＝5²＋8²－2×5×8×cos60°＝25＋64－80×(1／2)＝89－40＝49。よってa＝7。",
      "entities": [
        "law-of-cosines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-loc-cosa-from-567",
      "type": "input",
      "prompt": "b＝5、c＝8、a＝7の三角形で、cosAの値を求めよ。",
      "choices": null,
      "answer": "1/2",
      "accept": [
        "0.5"
      ],
      "explanation": "cosA＝(b²＋c²－a²)／(2bc)＝(25＋64－49)／80＝40／80＝1／2。",
      "entities": [
        "law-of-cosines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-loc-right-angle-reduces-choice",
      "type": "choice",
      "prompt": "余弦定理でA＝90°のとき、a²＝b²＋c²－2bc cosAはどうなるか。",
      "choices": [
        "a²＝b²－c²",
        "a²＝b²＋c²（三平方の定理）",
        "a²＝(b＋c)²",
        "a²＝2bc"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "A＝90°ではcosA＝0なので、－2bc cosAの項が0になり、a²＝b²＋c²（三平方の定理）になる。",
      "entities": [
        "law-of-cosines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-loc-triangle-type-5-6-7-choice",
      "type": "choice",
      "prompt": "a＝5、b＝6、c＝7の三角形は、どれにあたるか。",
      "choices": [
        "直角三角形",
        "鈍角三角形",
        "鋭角三角形",
        "三角形にならない"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "最も長い辺cと向かい合う角CについてcosC＝(25＋36－49)／60＝12／60＝0.2＞0。cosが正なのでCは鋭角。最大の角が鋭角なので、三角形全体も鋭角三角形。",
      "entities": [
        "law-of-cosines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-los-2r-from-a6-sin13",
      "type": "input",
      "prompt": "三角形ABCで、a＝6、sinA＝1／3のとき、外接円の直径2Rを求めよ。",
      "choices": null,
      "answer": "18",
      "accept": [],
      "explanation": "2R＝a／sinA＝6／(1／3)＝18。",
      "entities": [
        "law-of-sines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-los-b-from-b90-choice",
      "type": "choice",
      "prompt": "上と同じ三角形ABC（A＝30°、a＝4、2R＝8）でB＝90°のとき、辺bの長さはどれか。",
      "choices": [
        "4√2",
        "4",
        "4√3",
        "8"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "b／sinB＝2R＝8なので、b＝8×sin90°＝8×1＝8。",
      "entities": [
        "law-of-sines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-los-r-from-a4-a30",
      "type": "input",
      "prompt": "三角形ABCで、A＝30°、a＝4のとき、外接円の半径Rを求めよ。",
      "choices": null,
      "answer": "4",
      "accept": [],
      "explanation": "2R＝a／sinA＝4／sin30°＝4／(1／2)＝8。よってR＝4。",
      "entities": [
        "law-of-sines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-los-r-meaning-choice",
      "type": "choice",
      "prompt": "正弦定理a／sinA＝2RのRは、三角形の何にあたるか。",
      "choices": [
        "重心から頂点までの距離",
        "外心から辺までの距離",
        "内接円の半径",
        "外接円の半径"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "Rは三角形の3つの頂点をすべて通る外接円の半径。内心や重心の半径ではない。",
      "entities": [
        "law-of-sines"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-oa-cos120-choice",
      "type": "choice",
      "prompt": "120°のcosの値はどれか。",
      "choices": [
        "－√3／2",
        "－1／2",
        "1／2",
        "√3／2"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "120°＝180°－60°なので、cos120°＝－cos60°＝－1／2。符号を変え忘れると1／2と誤る。",
      "entities": [
        "obtuse-angle-trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-oa-sin-half-which-theta",
      "type": "choice",
      "prompt": "sinθ＝1／2（0°≤θ≤180°）を満たすθは次のうちどれか。",
      "choices": [
        "30°と150°",
        "30°のみ",
        "60°と120°",
        "150°のみ"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "sin30°＝1／2で成り立つ。sin(180°－30°)＝sin30°も成り立つので、θ＝30°と150°の両方が条件を満たす。",
      "entities": [
        "obtuse-angle-trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-oa-sin180",
      "type": "input",
      "prompt": "θ＝180°のとき、sinθの値を求めよ。",
      "choices": null,
      "answer": "0",
      "accept": [],
      "explanation": "θ＝180°では点Pが(－r,0)になるので、y＝0。sin180°＝0。",
      "entities": [
        "obtuse-angle-trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-oa-tan180",
      "type": "input",
      "prompt": "θ＝180°のとき、tanθの値を求めよ。",
      "choices": null,
      "answer": "0",
      "accept": [],
      "explanation": "θ＝180°では点Pが(－r,0)になるので、tan180°＝y／x＝0／(－r)＝0。",
      "entities": [
        "obtuse-angle-trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tam-area-6-8-30",
      "type": "input",
      "prompt": "辺b＝6、辺c＝8、間の角A＝30°の三角形の面積を求めよ。",
      "choices": null,
      "answer": "12",
      "accept": [],
      "explanation": "S＝(1／2)×6×8×sin30°＝(1／2)×48×(1／2)＝12。",
      "entities": [
        "triangle-area-and-measurement"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tam-area-8-10-150-choice",
      "type": "choice",
      "prompt": "2辺が8cmと10cmで、その間の角が150°の三角形の面積はどれか。",
      "choices": [
        "10cm²",
        "20cm²",
        "40cm²",
        "80cm²"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "S＝(1／2)×8×10×sin150°＝(1／2)×80×(1／2)＝20cm²。1／2をかけ忘れると40、sinの代わりに符号だけ変えるとマイナスの値になり誤る。",
      "entities": [
        "triangle-area-and-measurement"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tam-area-formula-why-choice",
      "type": "choice",
      "prompt": "面積の公式S＝(1／2)bc sinAは、どのように導かれるか。",
      "choices": [
        "三平方の定理を直接使う",
        "底辺×高さ÷2の高さをc sinAで表す",
        "余弦定理の両辺を2で割る",
        "正弦定理から直接求まる"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "底辺×高さ÷2の「高さ」の部分を、辺cと角Aからc sinAと表したもの。三平方の定理や正弦定理、余弦定理の式を直接変形したものではない。",
      "entities": [
        "triangle-area-and-measurement"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tam-pond-distance-120",
      "type": "input",
      "prompt": "OA＝3m、OB＝5m、∠AOB＝120°のとき、池をはさんだ2点間の距離ABを求めよ。",
      "choices": null,
      "answer": "7",
      "accept": [],
      "explanation": "余弦定理よりAB²＝3²＋5²－2×3×5×cos120°＝34－30×(－1／2)＝34＋15＝49。よってAB＝7m。",
      "entities": [
        "triangle-area-and-measurement"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tr-elevation-angle-height",
      "type": "input",
      "prompt": "観測地点から水平に30m離れた電波塔を見上げると、仰角が45°だった。電波塔の高さを求めよ。",
      "choices": null,
      "answer": "30",
      "accept": [],
      "explanation": "高さ＝距離×tan（仰角）＝30×tan45°＝30×1＝30m。",
      "entities": [
        "trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tr-sin-3-4-5",
      "type": "input",
      "prompt": "直角三角形ABC（∠C＝90°）で、AB＝5、BC＝3、AC＝4のとき、sinAの値を求めよ。",
      "choices": null,
      "answer": "3/5",
      "accept": [
        "0.6"
      ],
      "explanation": "sinA＝対辺／斜辺＝BC／AB＝3／5。",
      "entities": [
        "trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tr-tan-3-4-5",
      "type": "input",
      "prompt": "同じ直角三角形ABC（AB＝5、BC＝3、AC＝4）で、tanAの値を求めよ。",
      "choices": null,
      "answer": "3/4",
      "accept": [
        "0.75"
      ],
      "explanation": "tanA＝対辺／隣辺＝BC／AC＝3／4。",
      "entities": [
        "trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-tr-tan30-choice",
      "type": "choice",
      "prompt": "30°の正接（tan）の値はどれか。",
      "choices": [
        "√3／2",
        "√3",
        "√3／3",
        "1／2"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "tan30°＝√3／3。tan60°の√3や、sin30°の1／2と取り違えやすいので注意しよう。",
      "entities": [
        "trigonometric-ratios"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-trr-1-plus-tan2-from-cos",
      "type": "input",
      "prompt": "cosA＝1／3のとき、1＋tan²Aの値を求めよ。",
      "choices": null,
      "answer": "9",
      "accept": [],
      "explanation": "1＋tan²A＝1／cos²A＝1／(1／3)²＝1／(1／9)＝9。",
      "entities": [
        "trigonometric-ratio-relations"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-trr-cos-from-sin-3-5",
      "type": "choice",
      "prompt": "Aが鋭角でsinA＝3／5のとき、cosAの値はどれか。",
      "choices": [
        "3／5",
        "4／5",
        "5／4",
        "16／25"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "sin²A＋cos²A＝1より、cos²A＝1－9／25＝16／25。Aは鋭角でcosA＞0なので、cosA＝4／5。",
      "entities": [
        "trigonometric-ratio-relations"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-trr-sin70-as-cos",
      "type": "choice",
      "prompt": "sin70°をcosで表すとどれか。",
      "choices": [
        "cos110°",
        "cos70°",
        "cos20°",
        "sin20°"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "sinA＝cos(90°－A)より、sin70°＝cos(90°－70°)＝cos20°。",
      "entities": [
        "trigonometric-ratio-relations"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-trr-tan-from-sin-3-5",
      "type": "input",
      "prompt": "同じ条件（sinA＝3／5、cosA＝4／5）でtanAの値を求めよ。",
      "choices": null,
      "answer": "3/4",
      "accept": [
        "0.75"
      ],
      "explanation": "tanA＝sinA／cosA＝(3／5)／(4／5)＝3／4。",
      "entities": [
        "trigonometric-ratio-relations"
      ],
      "verified": true,
      "placeholder": false
    }
  ]
}