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Updated May 9, 2026

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正弦定理と余弦定理の基礎と活用

三角形の辺と角の関係を表す正弦定理と余弦定理は、図形問題を解くための最強ツール!共通試験でも頻出だから、公式を覚えるだけじゃなく、どっちを使うか瞬時に判断できるようになろう。

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# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

正弦定理・余弦定理の基本

三角形の計算で迷ったことない?正弦定理と余弦定理をマスターすれば、どんな三角形でも未知の辺や角を求められるようになる。

まず記号のルールから確認しよう。頂点をA、B、C、角の大きさもA、B、C、そして各角の対辺をそれぞれ小文字のa、b、cで表す。これを間違えると全部ずれちゃうから要注意!

外接円の半径Rも重要な要素だ。この記号の対応関係は絶対に崩しちゃダメ。

💡 覚え方のコツ
角は大文字、対辺は小文字で対応させる!A↔a、B↔b、C↔cの関係を体に染み込ませよう。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

正弦定理をマスターしよう

正弦定理の公式はこれ:asinA=bsinB=csinC=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

「辺とその対角のサインの比が一定」って覚えよう。しかもその比が外接円の直径2Rに等しいのがミソ!

使うタイミングは3つ。2角と1辺がわかっているとき、2辺と1対角がわかっているとき、そして外接円の半径が絡む問題のときだ。

ただし要注意!2辺と1対角から角を求める場合、解が2つ存在することがある。例えばsinB=12\sin B = \frac{1}{2}なら、B=30°か150°の可能性があるからね。

⚠️ 注意ポイント
正弦定理で角を求めるときは、鋭角と鈍角の2つの解がある可能性を常にチェック!

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

余弦定理で完璧に

余弦定理は三平方の定理の進化版!公式はa2=b2+c22bccosAa^2 = b^2 + c^2 - 2bc \cos Aだ。

角を求めたいときは変形してcosA=b2+c2a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}を使う。この形で覚えておくと計算が早いよ。

コサインの符号で角の種類がわかるのが便利!cosA>0\cos A > 0なら鋭角、cosA=0\cos A = 0なら直角、cosA<0\cos A < 0なら鈍角だ。

使うのは2辺とその間の角がわかっているときと、3辺全部がわかっているとき。余弦定理は解が一つに決まるから、正弦定理より扱いやすい場面も多い。

🎯 実践のコツ
余弦定理はコサインの符号で角の種類が即座にわかる!正弦定理よりシンプルで安心。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

実際に問題を解いてみよう

例題1:三角形ABCでb=6b=6A=30°A=30°C=45°C=45°のとき、辺ccと外接円の半径RRを求めよ。

まず残りの角B=180°30°45°=105°B = 180° - 30° - 45° = 105°を計算。2角と1辺がわかってるから正弦定理の出番だね!

正弦定理bsinB=csinC\frac{b}{\sin B} = \frac{c}{\sin C}を使って:c=6sin45°sin105°c = \frac{6 \sin 45°}{\sin 105°}

sin105°=sin(60°+45°)=6+24\sin 105° = \sin(60° + 45°) = \frac{\sqrt{6} + \sqrt{2}}{4}sin45°=22\sin 45° = \frac{\sqrt{2}}{2}を代入して計算すると、c=6(31)c = 6(\sqrt{3} - 1)になる。

外接円の半径は$2R = \frac{c}{\sin C}からからR = 362\sqrt{6} - \sqrt{2}$だ。

📝 計算のコツ
三角関数の値が複雑になったら、一番簡単な角を選んで計算しよう!

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

使い分けをマスターしよう

条件別の定理選択が試験のカギ!迷ったときのために整理しておこう。

余弦定理を使う場合

  • 2辺とその間の角 → 残りの1辺を求める
  • 3辺すべて → 各角を求める

正弦定理を使う場合

  • 2角と1辺 → 残りの辺を求める
  • 2辺とその対角 → 他の角を求める(解が2つの可能性あり)
  • 外接円の半径が絡む → 何でも求められる

鈍角の判断も重要だ。正弦定理では解が2つになる可能性があるけど、余弦定理ならcos\cosの符号で角の種類が即座にわかる。

🏆 最終チェック
「3辺と1角」なら余弦定理、「2辺2角」なら正弦定理!外接円が出たら即正弦定理を連想しよう。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

完璧な仕上げのために

面積公式S=12absinCS = \frac{1}{2}ab \sin Cも定理とセットで覚えておこう。角がわかれば面積も一発で求められる!

最重要ポイントをもう一度確認:

  • 正弦定理asinA=bsinB=csinC=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
  • 余弦定理a2=b2+c22bccosAa^2 = b^2 + c^2 - 2bc \cos A
  • 角を求めるときは鈍角の可能性も考慮

これらの定理は図形と計量の基本中の基本だ。どんな複雑な問題も、最終的には三角形に分割してこの定理を使うことが多い。

練習問題をたくさん解いて、条件を見た瞬間にどちらを使うべきか判断できるようになろう。君ならできる!

💪 励ましメッセージ
最初は迷うのが普通!問題をたくさん解けば、必ず瞬時に判断できるようになるよ。



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Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

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THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

 

共通試験

23

Updated May 9, 2026

6 pages

正弦定理と余弦定理の基礎と活用

三角形の辺と角の関係を表す正弦定理と余弦定理は、図形問題を解くための最強ツール!共通試験でも頻出だから、公式を覚えるだけじゃなく、どっちを使うか瞬時に判断できるようになろう。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

正弦定理・余弦定理の基本

三角形の計算で迷ったことない?正弦定理と余弦定理をマスターすれば、どんな三角形でも未知の辺や角を求められるようになる。

まず記号のルールから確認しよう。頂点をA、B、C、角の大きさもA、B、C、そして各角の対辺をそれぞれ小文字のa、b、cで表す。これを間違えると全部ずれちゃうから要注意!

外接円の半径Rも重要な要素だ。この記号の対応関係は絶対に崩しちゃダメ。

💡 覚え方のコツ
角は大文字、対辺は小文字で対応させる!A↔a、B↔b、C↔cの関係を体に染み込ませよう。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

正弦定理をマスターしよう

正弦定理の公式はこれ:asinA=bsinB=csinC=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

「辺とその対角のサインの比が一定」って覚えよう。しかもその比が外接円の直径2Rに等しいのがミソ!

使うタイミングは3つ。2角と1辺がわかっているとき、2辺と1対角がわかっているとき、そして外接円の半径が絡む問題のときだ。

ただし要注意!2辺と1対角から角を求める場合、解が2つ存在することがある。例えばsinB=12\sin B = \frac{1}{2}なら、B=30°か150°の可能性があるからね。

⚠️ 注意ポイント
正弦定理で角を求めるときは、鋭角と鈍角の2つの解がある可能性を常にチェック!

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

余弦定理で完璧に

余弦定理は三平方の定理の進化版!公式はa2=b2+c22bccosAa^2 = b^2 + c^2 - 2bc \cos Aだ。

角を求めたいときは変形してcosA=b2+c2a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}を使う。この形で覚えておくと計算が早いよ。

コサインの符号で角の種類がわかるのが便利!cosA>0\cos A > 0なら鋭角、cosA=0\cos A = 0なら直角、cosA<0\cos A < 0なら鈍角だ。

使うのは2辺とその間の角がわかっているときと、3辺全部がわかっているとき。余弦定理は解が一つに決まるから、正弦定理より扱いやすい場面も多い。

🎯 実践のコツ
余弦定理はコサインの符号で角の種類が即座にわかる!正弦定理よりシンプルで安心。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

実際に問題を解いてみよう

例題1:三角形ABCでb=6b=6A=30°A=30°C=45°C=45°のとき、辺ccと外接円の半径RRを求めよ。

まず残りの角B=180°30°45°=105°B = 180° - 30° - 45° = 105°を計算。2角と1辺がわかってるから正弦定理の出番だね!

正弦定理bsinB=csinC\frac{b}{\sin B} = \frac{c}{\sin C}を使って:c=6sin45°sin105°c = \frac{6 \sin 45°}{\sin 105°}

sin105°=sin(60°+45°)=6+24\sin 105° = \sin(60° + 45°) = \frac{\sqrt{6} + \sqrt{2}}{4}sin45°=22\sin 45° = \frac{\sqrt{2}}{2}を代入して計算すると、c=6(31)c = 6(\sqrt{3} - 1)になる。

外接円の半径は$2R = \frac{c}{\sin C}からからR = 362\sqrt{6} - \sqrt{2}$だ。

📝 計算のコツ
三角関数の値が複雑になったら、一番簡単な角を選んで計算しよう!

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

使い分けをマスターしよう

条件別の定理選択が試験のカギ!迷ったときのために整理しておこう。

余弦定理を使う場合

  • 2辺とその間の角 → 残りの1辺を求める
  • 3辺すべて → 各角を求める

正弦定理を使う場合

  • 2角と1辺 → 残りの辺を求める
  • 2辺とその対角 → 他の角を求める(解が2つの可能性あり)
  • 外接円の半径が絡む → 何でも求められる

鈍角の判断も重要だ。正弦定理では解が2つになる可能性があるけど、余弦定理ならcos\cosの符号で角の種類が即座にわかる。

🏆 最終チェック
「3辺と1角」なら余弦定理、「2辺2角」なら正弦定理!外接円が出たら即正弦定理を連想しよう。

# 正弦定理・余弦定理

## 正弦定理と余弦定理の概要

三角形の辺の長さと角の大きさの関係を表す重要な定理。これらを使いこなせれ
ば、三角形の未知の辺や角を計算で求められるようになる。図形問題の計算で必
須のツール。特に共通試験では、図形の性質と組み合わせて出題されることが多

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完璧な仕上げのために

面積公式S=12absinCS = \frac{1}{2}ab \sin Cも定理とセットで覚えておこう。角がわかれば面積も一発で求められる!

最重要ポイントをもう一度確認:

  • 正弦定理asinA=bsinB=csinC=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
  • 余弦定理a2=b2+c22bccosAa^2 = b^2 + c^2 - 2bc \cos A
  • 角を求めるときは鈍角の可能性も考慮

これらの定理は図形と計量の基本中の基本だ。どんな複雑な問題も、最終的には三角形に分割してこの定理を使うことが多い。

練習問題をたくさん解いて、条件を見た瞬間にどちらを使うべきか判断できるようになろう。君ならできる!

💪 励ましメッセージ
最初は迷うのが普通!問題をたくさん解けば、必ず瞬時に判断できるようになるよ。

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

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