Trigonometric phase and parametric curves
| English | 中文 | Pinyin |
|---|---|---|
| phase angle/feɪz ˈæŋɡl/ | 相位角 | xiàng wèi jiǎo |
| parametric curve/ˌpærəˈmetrɪk kɜːv/ | 参数曲线 | cān shù qū xiàn |
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?
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?
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 ____.
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.
Which answer fits this case?
Distinguish amplitude, angular frequency, phase angle and horizontal shift
Phase angle φ and horizontal shift φ/ω are always the same number.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.