| English | 意味 |
|---|---|
| period/ˈpɪərɪəd/ | 周期 |
| inverse/ɪnˈvɜːs/ | 逆関数 |
| sine/saɪn/ | 正弦 |
When does the wave hit a target?
- "At what times is the tide exactly $3$ m?" becomes a trig equation.
- Because the wave repeats, the answer is not one time but a whole rhythm of times.
- Solving means finding every angle where the function equals a target value.
- The unit circle and the period together deliver them all.
Finding the first solution
- To solve $\sin x = 0.5$, first find one angle with the inverse 反函数: $\arcsin(0.5) = \tfrac{\pi}{6}$.
- That gives a single "starter" solution inside the restricted range.
- For cosine or tangent, use $\arccos$ or $\arctan$ the same way.
- This first angle is the seed from which all others grow.
$\sin x = 0.5$の最初の解を見つけるには…を使う
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
逆関数は1つの角度を与える：$\arcsin(0.5) = 30°$（$\tfrac{\pi}{6}$）。残りは対称性を使って求める。
Then find them all
- The sine 正弦 wave meets a horizontal target line once every cycle, forever.
- So a trig equation usually has infinitely many solutions.
- Use the symmetry of the graph to find the second solution within one period.
- Then add whole multiples of the period 周期 to generate the endless rest.
すべての交点は解である
y = sin x
正弦波は水平な目標線を何度も交わるため、三角方程式には無数の解がある。
実数全域において、$\sin x = 0.5$には…がある
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
波が線 $y = 0.5$ と周期ごとに一度ずつ、永遠に交わる — 無限に多くの解。
すべての解をリストするには、最初の解に ____ の整数倍を加える。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
正弦は繰り返すため、その周期（$2\pi$）の任意の倍数を加えると別の解が得られる。
三角関数方程式について正しい文をすべて選択せよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
正弦は $1$ で上限があるため、$\sin x = 5$ は解を持たない。残りの3つは正しい。
Inequalities
- A trig inequality like $\sin x > 0.5$ asks for the intervals where the wave is above a line.
- Solve the matching equation first to find the boundary angles.
- Then read off which arcs of the wave satisfy the inequality.
- The answer is a repeating set of intervals, one per period.
方程式 $\sin x = 2$ に解はない。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
正弦は $1$ を超えないので、$\sin x = 2$ が起こることは決してない — 解は存在しない。
When there is no solution
- Sine and cosine never leave $[-1, 1]$, so $\sin x = 2$ has no solution at all.
- Always check that the target value is reachable before solving.
- A tangent target, by contrast, is always reachable, since tangent is unbounded.
- Knowing each function's range saves you from chasing impossible answers.
Do not report just the calculator's one answer. $\arcsin(0.5)$ gives $\tfrac{\pi}{6}$, but $\sin x = 0.5$ also holds at $\tfrac{5\pi}{6}$ and at every $+2\pi$ beyond both. A trig equation almost always has more solutions than the inverse alone reveals.
Solve $\cos x = 0$ for all $x$.
- One solution: $\arccos(0) = \tfrac{\pi}{2}$.
- Cosine is also $0$ at $\tfrac{3\pi}{2}$ — that is $\tfrac{\pi}{2} + \pi$.
- So the full solution is $x = \tfrac{\pi}{2} + k\pi$ for every integer $k$.
To solve a trig equation, use the inverse for a first angle, then the graph's symmetry and the period to list every solution — often infinitely many. Check the target is within the function's range first (sine and cosine cannot exceed $1$). Inequalities give repeating intervals.