三角関数の位相とパラメータ曲線
導入済み| English | 意味 |
|---|---|
| phase angle/feɪz ˈæŋɡl/ | 位相角 |
| parametric curve/ˌpærəˈmetrɪk kɜːv/ | パラメータ曲線 |
A decision before an answer
- A periodic signal can start at a minimum while keeping the same period and amplitude as one that starts at a maximum. Its phase, rather than its size or frequency, changes.
- Your goal: Distinguish amplitude, angular frequency, phase angle and horizontal shift.
Read the relationship
- For y=A cos(ωx−φ) with A>0 and ω>0, amplitude is A, period is 2π/ω, and horizontal shift is φ/ω. The phase angle φ is measured inside the cosine argument; it is not itself the horizontal shift unless ω=1. Start with peak-to-peak spacing for the period and midline-to-peak distance for amplitude. To find phase, substitute a known point and check the direction of motion there. Equivalent phases differ by 2π.
- Eliminate a parameter while retaining its domain and tracing direction.
For y=2 cos(3x−π), what is the rightward horizontal shift?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Factor the argument as 3(x−π/3). The phase π is divided by angular frequency 3.
Use the defining rule
- For y=−2 cos(3x), rewrite the model with positive amplitude as 2 cos(3x−π). Its amplitude is 2, period 2π/3, and phase π; the equivalent right shift is π/3. Peaks occur where 3x−π is a multiple of 2π. A graph point at x=0 and y=0 alone cannot determine phase uniquely: slopes or another point distinguish the possible angles. Units and angle conventions matter; ordinary calculus trigonometric derivatives use radians.
- Use parametric derivatives without assuming a vertical tangent is stationary.
For x=cos³t,y=sin³t with t in [0,π/2], which set is traced?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
The parameter interval keeps both coordinates nonnegative. Elimination gives the astroid relation, but does not remove this quadrant restriction.
Check the conditions
- A parametric curve specifies x=x(t), y=y(t). Eliminating t describes a point set but can lose restrictions and direction. For x=cos³t, y=sin³t, real cube roots give |x|^(2/3)+|y|^(2/3)=1, an astroid with cusps on the axes. As t runs from 0 to 2π it starts at (1,0), passes (0,1) at π/2 and travels counterclockwise. Restricting t to [0,π/2] gives only the first-quadrant arc, not the entire implicit locus.
- Use parametric derivatives without assuming a vertical tangent is stationary.
For y=−2 cos(3x), minima include x=0 and maxima include x=π/3. The phase representation 2 cos(3x−π) gives the same values. For x=2 cos t,y=sin t, elimination gives x²/4+y²=1; at t=π/4, dx/dt=−√2 and dy/dt=√2/2, hence slope −1/2. The full parameter interval [0,2π] traces the ellipse once counterclockwise.
For x=t²,y=t³ and t=2, the slope dy/dx is ____.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
dy/dx=(3t²)/(2t)=3t/2 when t≠0; at t=2 this equals 3.
Apply the task format
- When dx/dt≠0, dy/dx=(dy/dt)/(dx/dt). A horizontal tangent generally requires dy/dt=0 and dx/dt≠0; a vertical tangent generally reverses those conditions. If both vanish, inspect a limit or the local expansion rather than taking 0/0 as a slope. For x=t²,y=t³ at t=0, the quotient for t≠0 is 3t/2 and tends to zero: the cusp has a horizontal tangent. A zero parameter velocity is not by itself a local maximum or minimum of y as a function of x.
- Use parametric derivatives without assuming a vertical tangent is stationary.
Phase angle and horizontal shift differ by the frequency factor. An implicit equation can add untraced parts of a curve. If both parameter derivatives vanish, use local reasoning instead of labelling every such point an extremum.
このケースに当てはまる答えはどれか？
振幅、角周波数、位相角および水平シフトを区別する
Phase angle φ and horizontal shift φ/ω are always the same number.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
They coincide numerically only in special cases, such as ω=1. Frequency rescales the horizontal coordinate.
Keep the distinctions
- phase angle 相位角 — The angle offset inside a periodic function argument, defined modulo a full period.
- parametric curve 参数曲线 — A curve whose coordinates are specified as functions of a shared parameter.
- Distinguish amplitude, angular frequency, phase angle and horizontal shift.
- Eliminate a parameter while retaining its domain and tracing direction.
- Use parametric derivatives without assuming a vertical tangent is stationary.
Match each term with its precise meaning in this lesson.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.