Eliminate the parameter in the parametric equations x=1+sin t, y=3+2 sin t, for 0≤t≤π/2, and describe the curve, indicating its positive orientation. How does this curve differ from the curve x=1+sin t, y=3+2 sin t, for π/2≤t≤π?

검증된 단계별 안내

Start by writing down the given parametric equations: \(x = 1 + \sin t\) and \(y = 3 + 2 \sin t\), with the parameter \(t\) in the interval \(0 \leq t \leq \frac{\pi}{2}\).

Express \(\sin t\) from the first equation: \(\sin t = x - 1\).

Substitute \(\sin t = x - 1\) into the second equation to eliminate the parameter \(t\): \(y = 3 + 2(x - 1)\).

Simplify the resulting equation to get the Cartesian form of the curve: \(y = 3 + 2x - 2\) which simplifies to \(y = 2x + 1\).

To describe the curve's positive orientation, consider the interval \(0 \leq t \leq \frac{\pi}{2}\) where \(\sin t\) increases from 0 to 1, so \(x\) increases from 1 to 2 and \(y\) increases from 3 to 5. For the interval \(\frac{\pi}{2} \leq t \leq \pi\), \(\sin t\) decreases from 1 back to 0, so the curve is traced in the opposite direction, moving from \((2,5)\) back to \((1,3)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Eliminating the Parameter in Parametric Equations

Eliminating the parameter involves expressing one variable directly in terms of the other, removing the parameter (t). This is done by solving one parametric equation for the parameter and substituting into the other, resulting in a Cartesian equation that describes the curve.

Understanding the Domain and Orientation of Parametric Curves

The domain of the parameter t determines which portion of the curve is traced. Orientation refers to the direction in which the curve is drawn as t increases. Different intervals of t can produce different segments or directions on the same curve.

Analyzing Curve Differences Based on Parameter Intervals

Changing the interval of t can alter the part of the curve traced and its orientation. Comparing curves for different t-intervals helps identify how the shape, position, and direction of traversal differ, even if the parametric formulas are similar.