Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

검증된 단계별 안내

Start with the given parametric equations: \(x = 1 + \sin t\) and \(y = \cos t - 2\), where \(0 \leq t \leq \pi\).

Isolate the trigonometric functions from the parametric equations: from \(x = 1 + \sin t\), we get \(\sin t = x - 1\); from \(y = \cos t - 2\), we get \(\cos t = y + 2\).

Recall the Pythagorean identity for sine and cosine: \(\sin^2 t + \cos^2 t = 1\).

Substitute the expressions for \(\sin t\) and \(\cos t\) into the identity: \((x - 1)^2 + (y + 2)^2 = 1\).

This equation represents the Cartesian form of the particle's path, which is a circle centered at \((1, -2)\) with radius 1. The parameter interval \(0 \leq t \leq \pi\) indicates the particle moves along the upper half of this circle, starting at \(t=0\) and ending at \(t=\pi\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex paths and motions in the plane.

Eliminating the Parameter to Find Cartesian Equations

To find a Cartesian equation from parametric equations, the parameter t is eliminated by using algebraic manipulation or trigonometric identities. This process converts the parametric form into a single equation relating x and y, describing the particle’s path in the xy-plane.

Graphing and Interpreting Parametric Curves

Graphing parametric curves involves plotting points (x(t), y(t)) over the given parameter interval. Understanding the direction of motion and the portion of the curve traced requires analyzing how x and y change as t increases, which helps visualize the particle’s trajectory and orientation.