**Title: Parametric Equations in the xy-plane****Introduction:**Learn how to derive a Cartesian equation from parametric equations and understand the path of a particle in the xy-plane through a given parameter interval.**Problem Statement:**The parametric equations and parameter interval for the motion of a particle in the xy-plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.\[ x = \cos \left( \frac{5\pi}{4} - t \right), \quad y = \sin \left( \frac{5\pi}{4} - t \right), \quad 0 \leq t \leq \frac{\pi}{2} \]**Step-by-Step Solution:**1. **Express the Parametric Equations:** Provide the equations describing x and y in terms of t: \[ x = \cos \left( \frac{5\pi}{4} - t \right) \] \[ y = \sin \left( \frac{5\pi}{4} - t \right) \]2. **Graphing the Cartesian Equation:** To graph the Cartesian equation, we need to eliminate the parameter t. Applying trigonometric identities can help transform the parametric equations to a single Cartesian equation.3. **Convert to Cartesian Form:** \[ x = \cos \left( \frac{5\pi}{4} - t \right) \] Using the trigonometric identity $\cos (A - B) = \cos A \cos B + \sin A \sin B$: \[ x = \cos \frac{5\pi}{4} \cos t + \sin \frac{5\pi}{4} \sin t \] Similarly for y: \[ y = \sin \left( \frac{5\pi}{4} - t \right) \] Using the identity $\sin (A - B) = \sin A \cos B - \cos A \sin B$: \[ y = \sin \frac{5\pi}{4} \cos t - \cos \frac{5\pi}{4} \sin t \]4. **Simplify:**

If , then the independent variable x is called the input of the function and the dependent variable…

The parametric equations and parameter interval for the motion of a particle in the xy-plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. 5n y = sin 0sts5 X= cos The Cartesian equation for the particle is

The parametric equations and parameter interval for the motion of a particle in the xy-plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. 5n y = sin 0sts5 X= cos The Cartesian equation for the particle is

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.

Question

**Title: Parametric Equations in the xy-plane**

**Introduction:**

Learn how to derive a Cartesian equation from parametric equations and understand the path of a particle in the xy-plane through a given parameter interval.

**Problem Statement:**

The parametric equations and parameter interval for the motion of a particle in the xy-plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.

\[ x = \cos \left( \frac{5\pi}{4} - t \right), \quad y = \sin \left( \frac{5\pi}{4} - t \right), \quad 0 \leq t \leq \frac{\pi}{2} \]

**Step-by-Step Solution:**

1. **Express the Parametric Equations:**
Provide the equations describing x and y in terms of t:
\[ x = \cos \left( \frac{5\pi}{4} - t \right) \]
\[ y = \sin \left( \frac{5\pi}{4} - t \right) \]

2. **Graphing the Cartesian Equation:**
To graph the Cartesian equation, we need to eliminate the parameter t. Applying trigonometric identities can help transform the parametric equations to a single Cartesian equation.

3. **Convert to Cartesian Form:**
\[ x = \cos \left( \frac{5\pi}{4} - t \right) \]
Using the trigonometric identity $\cos (A - B) = \cos A \cos B + \sin A \sin B$:
\[ x = \cos \frac{5\pi}{4} \cos t + \sin \frac{5\pi}{4} \sin t \]

Similarly for y:
\[ y = \sin \left( \frac{5\pi}{4} - t \right) \]
Using the identity $\sin (A - B) = \sin A \cos B - \cos A \sin B$:
\[ y = \sin \frac{5\pi}{4} \cos t - \cos \frac{5\pi}{4} \sin t \]

4. **Simplify:**

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