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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 6
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Chapter 5, Problem 6

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 2 + 3 cos t, y = 4 + 2 sin t; t = π

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Identify the parametric equations given: \( x = 2 + 3 \cos t \) and \( y = 4 + 2 \sin t \).
Substitute the given value of \( t = \pi \) into the equation for \( x \): \( x = 2 + 3 \cos(\pi) \).
Calculate \( \cos(\pi) \), which is \(-1\), and substitute it into the equation: \( x = 2 + 3(-1) \).
Substitute the given value of \( t = \pi \) into the equation for \( y \): \( y = 4 + 2 \sin(\pi) \).
Calculate \( \sin(\pi) \), which is \(0\), and substitute it into the equation: \( y = 4 + 2(0) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically denoted as 't'. In this case, x and y are defined in terms of the parameter t, allowing for the representation of curves that may not be easily described by a single equation. Understanding how to evaluate these equations at specific values of t is crucial for finding points on the curve.
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Parameterizing Equations

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in relating angles to the ratios of sides in right triangles. In the given parametric equations, cos(t) and sin(t) are used to determine the x and y coordinates, respectively. Familiarity with the values of these functions at key angles, such as π, is essential for accurately calculating the coordinates of points on the curve.
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Introduction to Trigonometric Functions

Coordinate System

A coordinate system provides a framework for locating points in a plane using pairs of numbers (x, y). In this context, the x and y values derived from the parametric equations correspond to specific points on the Cartesian plane. Understanding how to plot these points and interpret their significance in relation to the curve is vital for visualizing the geometric representation of the equations.
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Intro to Polar Coordinates
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Related Practice
Textbook Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.


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