14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise

검증된 단계별 안내

Recall that the standard parametric equations for a circle centered at the origin with radius \(r\) are \(x = r \cos(t)\) and \(y = r \sin(t)\), where \(t\) is the parameter representing the angle in radians.

Since the given circle is \(x^2 + y^2 = 9\), the radius \(r\) is \(3\). So the standard parametric form is \(x = 3 \cos(t)\) and \(y = 3 \sin(t)\).

Note that the standard parametric form traces the circle counterclockwise as \(t\) increases. To generate the circle clockwise, we need to reverse the direction of traversal.

To reverse the direction, replace \(t\) by \(-t\) in the parametric equations. This gives \(x = 3 \cos(-t)\) and \(y = 3 \sin(-t)\).

Use the even and odd properties of cosine and sine: \(\cos(-t) = \cos(t)\) and \(\sin(-t) = -\sin(t)\). So the parametric equations for the circle traced clockwise are \(x = 3 \cos(t)\) and \(y = -3 \sin(t)\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing more flexible descriptions of curves, including direction and speed.

Equation of a Circle

The standard equation x² + y² = r² represents a circle centered at the origin with radius r. For r = 3, the circle includes all points (x, y) satisfying x² + y² = 9, forming a closed curve equidistant from the center.

Orientation and Direction in Parametric Curves

Parametric curves have an orientation determined by how the parameter t increases. For a circle, the direction can be counterclockwise or clockwise, which affects the sign and order of the parametric functions, such as using sine and cosine with positive or negative signs.

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.The circle x ² + y ² =9, generated clockwise