| English | 意味 |
|---|---|
| sine/saɪn/ | 正弦 |
| cosine/ˈkəʊsaɪn/ | 余弦 |
| tangent/ˈtændʒənt/ | 正切 |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | 単位円 |
| radians/ˈreɪdɪənz/ | ラジアン |
Three ratios that unlock circles
- Long ago, surveyors and astronomers needed to link angles to lengths.
- The answer was three ratios inside a right triangle: sine, cosine, and tangent.
- They turn an angle into a number, and a number back into an angle.
- These three functions are the doorway to everything periodic.
Right-triangle ratios
- For an acute angle in a right triangle, label the sides opposite, adjacent, and hypotenuse.
- Sine 正弦: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$.
- Cosine 余弦: $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$.
- Tangent 正切: $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$. Remember SOH-CAH-TOA.
直角三角形において、$\sin\theta$ は…に等しい
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
SOH: sine は対辺÷斜辺。CAH: cosine は隣辺÷斜辺。TOA: tangent は対辺÷隣辺。
The unit circle
- Draw a circle of radius $1$ centred at the origin — the unit circle 单位圆.
- A point on it at angle $\theta$ has coordinates exactly $(\cos\theta, \sin\theta)$.
- So cosine is the x-coordinate and sine is the y-coordinate.
- This lets angles beyond $90°$ have sine and cosine too — all the way around.
単位円周上で点を回転させる
点のx座標は角度のcosineであり、y座標はsineである。
単位円周上、角度$\theta$ の点の座標は…である
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
半径1 の円において、点は$(\cos\theta, \sin\theta)$ — cosineはx座標、sineはy座標である。
正しい記述をすべて select せよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
全周は$360°$（または$2\pi$ラジアン）であり、$100°$ではない。残りの三つは正しい。
Radians and tangent
- Angles can be measured in degrees or in radians 弧度, where a full circle is $2\pi$.
- Radians measure the arc length swept on the unit circle, so $180° = \pi$.
- Tangent connects the other two: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
- On the unit circle, tangent is the slope of the line from the origin to the point.
tangentはsineを____で割ったもの（つまり$\tan\theta = \tfrac{\sin\theta}{\cos\theta}$）。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
tangentはsineをcosineで割ったもの：$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$。
角度はラジアンで測ることもでき、全周は$2\pi$である。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
全周は$360°$または、ラジアンでは$2\pi$である。ラジアンは単位円周上の弧の長さを表す。
Around and around
- As the point travels once around the circle, sine and cosine cycle through all their values.
- Past one full turn, everything repeats — which is why these functions are periodic.
- Cosine starts at $1$ (angle $0$); sine starts at $0$.
- The unit circle is the single picture behind every trig value.
Keep track of your angle units. $\sin(90)$ is nearly $1$ if $90$ means degrees, but a very different value if $90$ means radians. Always know whether a problem is in degrees or radians before you compute.
Find $\sin\theta$ and $\cos\theta$ for the angle whose unit-circle point is $(0.6, 0.8)$.
- The x-coordinate is cosine: $\cos\theta = 0.6$.
- The y-coordinate is sine: $\sin\theta = 0.8$.
- Check: $0.6^2 + 0.8^2 = 0.36 + 0.64 = 1$, as the unit circle requires.
Sine, cosine, and tangent are ratios in a right triangle (SOH-CAH-TOA). On the unit circle, a point at angle $\theta$ is $(\cos\theta, \sin\theta)$, which extends the functions to any angle. Angles measure in degrees or radians ($2\pi$ per turn), and $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$.