[{"data":1,"prerenderedAt":170},["ShallowReactive",2],{"topic-page:\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions\u002Funit-circle-trigonometric-values":3},{"topic":4,"category":139,"seo":143,"breadcrumbs":148,"prerequisites":155,"nextTopics":163,"relatedTopics":169},{"id":5,"slug":6,"title":7,"summary":8,"canonicalPath":9,"categoryPath":10,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"generatedAt":14,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":16,"expectedKnowledge":21,"learningGoal":25,"blocks":26},"high-school-mathematics-ii-trigonometric-functions-unit-circle-trigonometric-values","unit-circle-trigonometric-values","単位円と三角関数の値","一般角の終辺と単位円の交点からsin、cos、tanを定義し、基本角の値と象限ごとの符号を判断します。","\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions\u002Funit-circle-trigonometric-values","\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions","beginner","student_high",6,"2026-07-27","ai_generated",[17,18,19,20],"数学II","三角関数","単位円","基本値",[22,23,24],"一般角","弧度法","数学Iの三角比","一般角の終辺と単位円の交点からsin、cos、tanを定義し、基本角の値と象限ごとの符号を判断できるようになる。",[27,32,42,66,72,82,92,98,104,120,129],{"id":28,"type":29,"title":7,"subtitle":30,"shortDefinition":31},"hero","HeroBlock","終点の座標がsinとcos","半径1の円を使うと、どんな一般角でも三角比を座標として定義できます。終点の横座標がcos、縦座標がsin、縦を横で割った値がtanです。",{"id":33,"type":34,"term":35,"definition":36,"plainExplanation":37,"keywords":38},"definition","DefinitionBlock","単位円による三角関数","一般角θの終辺と単位円の交点Pの座標を用いて定めるsin、cos、tanです。","P=(x,y)とするとcosθ=x、sinθ=y、tanθ=y\u002Fxです。第2・第3象限ではxが負、第3・第4象限ではyが負になるため、値の符号を図から判断できます。",[39,40,41],"cosはx座標","sinはy座標","tanはy\u002Fx",{"id":43,"type":44,"title":45,"formulas":46,"strategy":65},"formula","FormulaBlock","単位円上の座標定義",[47],{"label":48,"expression":49,"meaning":50,"symbols":51,"tip":64},"終点P","P(cos θ, sin θ),  tan θ=sin θ\u002Fcos θ","単位円上の終点の横座標がcos、縦座標がsinで、cosが0でないとき比がtan。",[52,55,58,61],{"symbol":53,"meaning":54},"θ","始辺から終辺までの一般角",{"symbol":56,"meaning":57},"P","終辺と単位円の交点",{"symbol":59,"meaning":60},"cos θ","Pのx座標",{"symbol":62,"meaning":63},"sin θ","Pのy座標","tanは横座標が0の角では定義されない","基準角の大きさを求め、象限からx座標とy座標の符号を付けます。",{"id":67,"type":68,"src":69,"alt":70,"caption":71},"diagram","DiagramBlock","\u002Fassets\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions\u002Funit-circle-trigonometric-values\u002Funit-circle-trigonometric-values.svg","第2象限の単位円上の点からx座標cosθとy座標sinθを軸へ投影する図","第2象限の終点Pからx軸・y軸へ破線を下ろし、横座標cosθが負、縦座標sinθが正になることを示します。回転矢印と基準角も併記します。",{"id":73,"type":74,"steps":75,"title":81},"steps","StepBlock",[76,77,78,79,80],"角を0以上2π未満の同じ終辺の角へ直す","終辺がどの象限にあるか決める","x軸との鋭角である基準角を求める","基本角の絶対値を使う","象限からsin、cos、tanの符号を付ける","値を求める手順",{"id":83,"type":84,"title":85,"points":86,"body":91},"key-points","KeyPointBlock","象限ごとの符号",[87,88,89,90],"第1象限はすべて正","第2象限はsinが正","第3象限はtanが正","第4象限はcosが正","第1象限ではsin、cos、tanがすべて正です。第2象限ではsinだけ正、第3象限ではtanだけ正、第4象限ではcosだけ正です。座標の符号から毎回導けます。",{"id":93,"type":94,"scenario":95,"explanation":96,"result":97},"example","ExampleBlock","θ=5π\u002F6のsin、cos、tanを求める。","5π\u002F6は第2象限で、基準角はπ\u002F6です。π\u002F6の基本値から絶対値はsinが1\u002F2、cosが√3\u002F2、tanが1\u002F√3です。第2象限なのでsinは正、cosとtanは負です。","sinθ=1\u002F2、cosθ=−√3\u002F2、tanθ=−1\u002F√3です。",{"id":99,"type":100,"warningTitle":101,"message":102,"severity":99,"action":103},"warning","WarningBlock","sinとcosの座標を逆にしない","単位円の横方向がcos、縦方向がsinです。数学Iの直角三角形で向かい側・隣り側だけを暗記すると混同しやすくなります。","終点をP(cosθ,sinθ)と座標の順に書いてから値を読みましょう。",{"id":105,"type":106,"title":107,"questions":108},"quiz","QuizBlock","確認テスト",[109],{"question":110,"choices":111,"answerIndex":115,"choiceExplanations":116},"第3象限に終辺がある角θについて、三角関数の符号の組合せとして正しいものはどれですか。",[112,113,114],"sinθ\u003C0、cosθ\u003C0、tanθ>0","sinθ>0、cosθ\u003C0、tanθ\u003C0","sinθ\u003C0、cosθ>0、tanθ\u003C0",0,[117,118,119],"正解です。第3象限ではxもyも負なので、その比tanは正です。","これは第2象限の符号です。第3象限ではy座標も負です。","これは第4象限の符号です。第3象限ではx座標も負です。",{"id":121,"type":122,"title":123,"points":124},"summary","SummaryBlock","まとめ",[125,126,127,128],"単位円の終点はP(cosθ,sinθ)","tanθはsinθ\u002Fcosθで、cosθ=0では未定義","基準角で絶対値、象限で符号を決める","一般角の三角関数は数学Iの三角比を自然に拡張する",{"id":130,"type":131,"title":132,"nextTopics":133,"relatedKeywords":135},"next-action","NextActionBlock","単位円から基本恒等式を導く",[134],"high-school-mathematics-ii-trigonometric-functions-trigonometric-identities-and-properties",[136,137,138],"相互関係","恒等式","象限",{"path":10,"title":18,"description":140,"parentPath":141,"accentToken":142},"一般角、弧度法、三角関数のグラフ、相互関係、加法定理を整理していくカテゴリです。","\u002Fhigh-school\u002Fmathematics-ii","signal-blue",{"title":144,"description":145,"canonicalUrl":146,"ogImage":147},"単位円と三角関数の値 | 数学II 三角関数","単位円上の座標をcosとsinに対応させ、象限と基本角から三角関数の値を求めます。","https:\u002F\u002Fzeqnilo.com\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions\u002Funit-circle-trigonometric-values","https:\u002F\u002Fzeqnilo.com\u002Fassets\u002Fsite\u002Fdefault-og.svg",[149,152,153,154],{"title":150,"path":151},"高校","\u002Fhigh-school",{"title":17,"path":141},{"title":18,"path":10},{"title":7,"path":9},[156],{"id":157,"title":158,"summary":159,"canonicalPath":160,"categoryPath":10,"categoryTitle":18,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":161},"high-school-mathematics-ii-trigonometric-functions-radians-arcs-and-sectors","弧度法と扇形","弧度法を半径に対する弧長の比として理解し、度数法との変換と扇形の弧長・面積を一つの関係で求めます。","\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions\u002Fradians-arcs-and-sectors",[17,18,23,162],"扇形",[164],{"id":134,"title":165,"summary":166,"canonicalPath":167,"categoryPath":10,"categoryTitle":18,"difficulty":11,"targetAudience":12,"estimatedMinutes":13,"publishedAt":14,"updatedAt":14,"reviewStatus":15,"tags":168},"三角関数の相互関係","単位円から三角関数の基本恒等式を理解し、一つの値と象限の条件から他の三角関数の値を求めます。","\u002Fhigh-school\u002Fmathematics-ii\u002Ftrigonometric-functions\u002Ftrigonometric-identities-and-properties",[17,18,136,137],[],1785565324768]