Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Eliminate a parameter with an identity.
- Determine traversal direction.
- Calculate velocity magnitude at two times.
Before you start
Trig identities and derivative rules d(cos t)/dt = −sin t and d(sin t)/dt = cos t.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A point follows x(t) = 3cos t and y(t) = 2sin t metres for 0 ≤ t ≤ 2π seconds, with phase increasing at 1 rad/s. Find the curve, travel direction, and speeds at t = 0 and π/2.
Why this math matters
Parametric equations say how a curve is traced, not just which points belong to it. Equal phase increments need not produce equal distances along a noncircular path.
Set up the model
A useful answer starts with clear assumptions:
- Coordinates are orthonormal and lengths are in metres.
- The numerical phase in radians equals the numerical time in seconds.
- The stated formulas describe an ideal motion exactly.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Trace an ellipse and compare its changing speed
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A point follows x(t) = 3cos t and y(t) = 2sin t metres for 0 ≤ t ≤ 2π seconds, with phase increasing at 1 rad/s. Find the curve, travel direction, and speeds at t = 0 and π/2.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Identify the curve
x²/9 + y²/4 = cos²t + sin²t = 1
The coordinate relation is an ellipse with horizontal semiaxis 3 and vertical semiaxis 2.
Follow successive positions
t = 0: (3, 0); t = π/2: (0, 2); t = π: (−3, 0)
Starting at the rightmost point and moving toward the top establishes counterclockwise traversal.
Compare velocities
v(t) = (−3sin t, 2cos t); |v(0)| = 2 m/s; |v(π/2)| = 3 m/s
Speed is the vector magnitude. At each selected time one component vanishes, making the comparison immediate.
The result
The point moves counterclockwise on x²/9 + y²/4 = 1; its two speeds are 2 m/s and 3 m/s.
The period is 2π seconds, but constant phase rate does not imply constant speed around this ellipse.
Common mistakes to catch
- A curve equation alone omits direction and timing.
- Adding signed velocity components does not calculate speed.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For x = 4cos t, y = 4sin t, find the speed.
Show a hint
Use sin²t + cos²t = 1.
Reveal answer and explanation
4 units per unit time
√(16sin²t + 16cos²t) = 4 at every time.
Practice 2
Replace y(t) by −2sin t in the original motion. Which direction does it trace?
Show a hint
Check where it goes immediately after (3, 0).
Reveal answer and explanation
Clockwise
The point initially moves downward from the rightmost point.
Take the idea with you
Keep the parameter's units and interval when eliminating it from a coordinate model.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Build a new waveform from three sine waves
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