31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=2 sin 8t, y=2 cos 8t

검증된 단계별 안내

Start with the given parametric equations: \(x = 2 \sin(8t)\) and \(y = 2 \cos(8t)\).

Isolate the trigonometric functions by dividing both \(x\) and \(y\) by 2: \(\frac{x}{2} = \sin(8t)\) and \(\frac{y}{2} = \cos(8t)\).

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

Substitute \(\sin(8t)\) and \(\cos(8t)\) with \(\frac{x}{2}\) and \(\frac{y}{2}\) respectively in the identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1\).

Simplify the equation to get a single equation in terms of \(x\) and \(y\): \(\frac{x^2}{4} + \frac{y^2}{4} = 1\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Parametric Equations

Parametric equations express the coordinates x and y as functions of a third variable, usually t, called the parameter. Understanding how x and y depend on t allows us to describe curves that may not be functions in the traditional sense.

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the variable t, resulting in a single equation relating x and y. This often requires using trigonometric identities or algebraic techniques to combine the expressions.

Trigonometric Identities

Trigonometric identities, such as sin²θ + cos²θ = 1, are essential tools for eliminating parameters when x and y are defined using sine and cosine functions. Applying these identities helps convert parametric forms into standard Cartesian equations.

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.x=2 sin 8t, y=2 cos 8t