Chapter 14: Q. 15 (page 1106)

Give a smooth parametrization $r (t)$ for the unit circle, starting and ending at (1, 0) and travelling in the counterclockwise direction.

Short Answer

Expert verified

Aparametrization $r (t)$ for the unit circle $r (t) = \cos t i + \sin t j, 0 \leq t \leq 2 π$ .

Step by step solution

Step 1. Given Information

Give a smooth parametrization $r (t)$ for the unit circle, starting and ending at (1, 0) and travelling in the counterclockwise direction.

Step 2. A parametric curve is defined as a collection of points given by two continuous functions x(t) and y(t), that is, the points on the curve are the collection of points (x(t), y(t)) where x and y are continuous functions of t.

We wish to parameterize the unit circle.

The unit circle is defined by the equation $x^{2} + y^{2} = 1 .$

From elementary trigonometry we recall the identity $(c o s (t))^{2} + (s i n (t))^{2} = 1$ for all $[0, 2 π)$ .

This directly gives us our first parametrization of the unit circle;

$r (t) = \cos t i + \sin t j, 0 \leq t \leq 2 π$

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Give a smooth parametrization $r (t)$ for the unit circle, starting and ending at (1, 0) and travelling in the counterclockwise direction.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. A parametric curve is defined as a collection of points given by two continuous functions x(t) and y(t), that is, the points on the curve are the collection of points (x(t), y(t)) where x and y are continuous functions of t.

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Give a smooth parametrization r(t) for the unit circle, starting and ending at (1, 0) and travelling in the counterclockwise direction.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. A parametric curve is defined as a collection of points given by two continuous functions x(t) and y(t), that is, the points on the curve are the collection of points (x(t), y(t)) where x and y are continuous functions of t.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Calculus

Geometry

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

Give a smooth parametrization $r (t)$ for the unit circle, starting and ending at (1, 0) and travelling in the counterclockwise direction.