How do I solve 42? **Topic: Converting Parametric Equations to Rectangular Form and Writing Parametric Equations****Objective 2: Eliminate the Parameter**Convert the parameter and write an equation in rectangular coordinates to describe the given curve and indicate its orientation._For Exercises 11-26:_1. Solve parametric equations. - Example 1: $ x = t + 2 $ and $ y = \frac{t}{t+1} $ - Example 2: $ x = 1 $ and $ y = t $ 2. Simplify complex expressions. - Example 3: $ x = r $ and $ y = \sqrt{1-t^2} $3. Apply trigonometric identities. - Example 4: $ x = \cos \theta $ and $ y = \sin^2 \theta - 4 $_For Exercises 27-30:_Eliminate the parameter and write an equation in rectangular coordinates to represent the given curves.1. Circles and Ellipses: - Example 5: Circle: $ x = h + r \cos \theta $ and $ y = k + r \sin \theta $2. Lines through two points: - Example 6: Line through points $(x_1, y_1)$ and $(x_2, y_2)$3. Hyperbolas: - Example 7: Hyperbola: $ x = h + a \sec \theta $ and $ y = k + b \tan \theta $**Objective 3: Write Parametric Equations to Represent a Curve**Write parametric equations using given definitions of $ x $._For Exercises 31-34:_1. Begin by simplifying given algebraic equations. - Example 8: $ x = t + 4, x > 0 $2. Derive from specific defined conditions. - Example 9: $ x = r, t \geq 0 $_For Exercises 35-42:_Use the results of Exercises 27-30 to write parametric equations to represent given curves.1. Lines passing through given points. - Example 10: Line through $(0, 2)$ and $(3, 1)$2. Circles and Ellipses by center, vertices, and

The center of the ellipse 0,0, vertices 0,±5 and foci 0,±4. We have to find the equation of the…

Answered: Ellipse with center (0, 0), vertices…

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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How do I solve 42?

**Topic: Converting Parametric Equations to Rectangular Form and Writing Parametric Equations**

**Objective 2: Eliminate the Parameter**

Convert the parameter and write an equation in rectangular coordinates to describe the given curve and indicate its orientation.

_For Exercises 11-26:_

1. Solve parametric equations.
- Example 1: $ x = t + 2 $ and $ y = \frac{t}{t+1} $
- Example 2: $ x = 1 $ and $ y = t $

2. Simplify complex expressions.
- Example 3: $ x = r $ and $ y = \sqrt{1-t^2} $

3. Apply trigonometric identities.
- Example 4: $ x = \cos \theta $ and $ y = \sin^2 \theta - 4 $

_For Exercises 27-30:_

Eliminate the parameter and write an equation in rectangular coordinates to represent the given curves.

1. Circles and Ellipses:
- Example 5: Circle: $ x = h + r \cos \theta $ and $ y = k + r \sin \theta $

2. Lines through two points:
- Example 6: Line through points $(x_1, y_1)$ and $(x_2, y_2)$

3. Hyperbolas:
- Example 7: Hyperbola: $ x = h + a \sec \theta $ and $ y = k + b \tan \theta $

**Objective 3: Write Parametric Equations to Represent a Curve**

Write parametric equations using given definitions of $ x $.

_For Exercises 31-34:_

1. Begin by simplifying given algebraic equations.
- Example 8: $ x = t + 4, x > 0 $

2. Derive from specific defined conditions.
- Example 9: $ x = r, t \geq 0 $

_For Exercises 35-42:_

Use the results of Exercises 27-30 to write parametric equations to represent given curves.

1. Lines passing through given points.
- Example 10: Line through $(0, 2)$ and $(3, 1)$

2. Circles and Ellipses by center, vertices, and

Expert Solution

Step 1: Given,

The center of the ellipse $(0, 0)$ , vertices $(0, \pm 5)$ and foci $(0, \pm 4) .$ We have to find the equation of the ellipse.

Step 2: Concept Used

The equation of an ellipse is $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$ for a horizontally oriented ellipse and $\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$ for a vertically oriented ellipse.

$(h, k)$ is the center and the distance c from the center to the foci is given by $a^{2} - b^{2} = c^{2} .$ a is the distance from the center to the vertices and b is the distance from the center to the co-vertices.

The center of the ellipse is halfway between the vertices. Thus, the center $(h, k)$ of the ellipse is $(0, 0)$ and the ellipse is vertically oriented.

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